Why is commutativity optional in multiplication for rings?Why is ring addition commutative?Why can't the...

What happens if a wizard reaches level 20 but has no 3rd-level spells that they can use with the Signature Spells feature?

Emit zero-width bash prompt sequence from external binary

Avoiding morning and evening handshakes

Do authors have to be politically correct in article-writing?

What's the purpose of these copper coils with resitors inside them in A Yamaha RX-V396RDS amplifier?

Could be quantum mechanics necessary to analyze some biology scenarios?

Why is working on the same position for more than 15 years not a red flag?

How to add multiple differently colored borders around a node?

On what did Lego base the appearance of the new Hogwarts minifigs?

Can a person refuse a presidential pardon?

Is my plan for fixing my water heater leak bad?

F1 visa even for a three-week course?

Obtaining a matrix of complex values from associations giving the real and imaginary parts of each element?

Which branches of mathematics can be done just in terms of morphisms and composition?

Eww, those bytes are gross

Why do neural networks need so many training examples to perform?

What is Crew Dragon approaching in this picture?

Am I a Rude Number?

Why does the DC-9-80 have this cusp in its fuselage?

How do we edit a novel that's written by several people?

Why is this code uniquely decodable?

How should I state my MS degree in my CV when it was in practice a joint-program?

Called into a meeting and told we are being made redundant (laid off) and "not to share outside". Can I tell my partner?

How can I mix up weapons for large groups of similar monsters/characters?



Why is commutativity optional in multiplication for rings?


Why is ring addition commutative?Why can't the Polynomial Ring be a Field?Why Does Induction Prove Multiplication is Commutative?In a Commutative Ring, is Addition Necessarily Commutative?Is there a theory of “rings” with partially defined multiplication?Why do they need commutativity in the proof?What is a semiring without commutativity requirement for addition?Does commutativity of $+$ on $mathbb{N}$ imply the associativity of $+$ on $mathbb{N}$?PID which is not a Prüfer domainAbout the hypotheses for a ring to contain a maximal idealAre rings closed under addition and multiplication?













4












$begingroup$


More precisely, why is it that all rings are required by the axioms to have commutativity in addition, but are not held to the same axiom regarding multiplication? I know that we have commutative and non-commutative rings depending on whether or not they are commutative in multiplication, but I am wondering why it is that the axioms were defined that way, providing us with this option.



I am using this list of axioms, from David Sharpe’s Rings and factorization:
Ring axioms










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
    $endgroup$
    – Eevee Trainer
    2 hours ago










  • $begingroup$
    @EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
    $endgroup$
    – Kusa
    2 hours ago






  • 1




    $begingroup$
    Note that matrices are an example where addition is commutative but multiplication is not necessarily
    $endgroup$
    – J. W. Tanner
    2 hours ago










  • $begingroup$
    Oh, I did miss it, sorry about that Kusa. xD
    $endgroup$
    – Eevee Trainer
    2 hours ago






  • 2




    $begingroup$
    The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
    $endgroup$
    – Arturo Magidin
    2 hours ago
















4












$begingroup$


More precisely, why is it that all rings are required by the axioms to have commutativity in addition, but are not held to the same axiom regarding multiplication? I know that we have commutative and non-commutative rings depending on whether or not they are commutative in multiplication, but I am wondering why it is that the axioms were defined that way, providing us with this option.



I am using this list of axioms, from David Sharpe’s Rings and factorization:
Ring axioms










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
    $endgroup$
    – Eevee Trainer
    2 hours ago










  • $begingroup$
    @EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
    $endgroup$
    – Kusa
    2 hours ago






  • 1




    $begingroup$
    Note that matrices are an example where addition is commutative but multiplication is not necessarily
    $endgroup$
    – J. W. Tanner
    2 hours ago










  • $begingroup$
    Oh, I did miss it, sorry about that Kusa. xD
    $endgroup$
    – Eevee Trainer
    2 hours ago






  • 2




    $begingroup$
    The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
    $endgroup$
    – Arturo Magidin
    2 hours ago














4












4








4


1



$begingroup$


More precisely, why is it that all rings are required by the axioms to have commutativity in addition, but are not held to the same axiom regarding multiplication? I know that we have commutative and non-commutative rings depending on whether or not they are commutative in multiplication, but I am wondering why it is that the axioms were defined that way, providing us with this option.



I am using this list of axioms, from David Sharpe’s Rings and factorization:
Ring axioms










share|cite|improve this question











$endgroup$




More precisely, why is it that all rings are required by the axioms to have commutativity in addition, but are not held to the same axiom regarding multiplication? I know that we have commutative and non-commutative rings depending on whether or not they are commutative in multiplication, but I am wondering why it is that the axioms were defined that way, providing us with this option.



I am using this list of axioms, from David Sharpe’s Rings and factorization:
Ring axioms







abstract-algebra ring-theory commutative-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









Eevee Trainer

7,05321337




7,05321337










asked 2 hours ago









KusaKusa

323




323








  • 1




    $begingroup$
    For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
    $endgroup$
    – Eevee Trainer
    2 hours ago










  • $begingroup$
    @EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
    $endgroup$
    – Kusa
    2 hours ago






  • 1




    $begingroup$
    Note that matrices are an example where addition is commutative but multiplication is not necessarily
    $endgroup$
    – J. W. Tanner
    2 hours ago










  • $begingroup$
    Oh, I did miss it, sorry about that Kusa. xD
    $endgroup$
    – Eevee Trainer
    2 hours ago






  • 2




    $begingroup$
    The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
    $endgroup$
    – Arturo Magidin
    2 hours ago














  • 1




    $begingroup$
    For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
    $endgroup$
    – Eevee Trainer
    2 hours ago










  • $begingroup$
    @EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
    $endgroup$
    – Kusa
    2 hours ago






  • 1




    $begingroup$
    Note that matrices are an example where addition is commutative but multiplication is not necessarily
    $endgroup$
    – J. W. Tanner
    2 hours ago










  • $begingroup$
    Oh, I did miss it, sorry about that Kusa. xD
    $endgroup$
    – Eevee Trainer
    2 hours ago






  • 2




    $begingroup$
    The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
    $endgroup$
    – Arturo Magidin
    2 hours ago








1




1




$begingroup$
For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
$endgroup$
– Eevee Trainer
2 hours ago




$begingroup$
For what it's worth, we also have the notion of a "commutative ring," one in which multiplication does commute. (en.wikipedia.org/wiki/Commutative_ring) Of course I get you're trying to get at why we don't require this, and I don't know how to answer you on that, but I figured it's a point worth bringing up.
$endgroup$
– Eevee Trainer
2 hours ago












$begingroup$
@EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
$endgroup$
– Kusa
2 hours ago




$begingroup$
@EeveeTrainer Thank you! I dunno if you missed it, but I did express my awareness of that.
$endgroup$
– Kusa
2 hours ago




1




1




$begingroup$
Note that matrices are an example where addition is commutative but multiplication is not necessarily
$endgroup$
– J. W. Tanner
2 hours ago




$begingroup$
Note that matrices are an example where addition is commutative but multiplication is not necessarily
$endgroup$
– J. W. Tanner
2 hours ago












$begingroup$
Oh, I did miss it, sorry about that Kusa. xD
$endgroup$
– Eevee Trainer
2 hours ago




$begingroup$
Oh, I did miss it, sorry about that Kusa. xD
$endgroup$
– Eevee Trainer
2 hours ago




2




2




$begingroup$
The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
$endgroup$
– Arturo Magidin
2 hours ago




$begingroup$
The first rings that were considered were generally commutative, but it soon became apparent that rings with noncommutative multiplication were far too common, starting with matrices and, more generally, endomorphism rings of abelian groups. On the other hand, if you take the definition of ring with unity, but omit the condition that addition is commutative, it turns out that you can prove that the other conditions force commutativity of addition.
$endgroup$
– Arturo Magidin
2 hours ago










1 Answer
1






active

oldest

votes


















5












$begingroup$

The first rings that were considered were generally commutative; polynomial rings, then from work of Dedekind in number fields. The properties were then abstracted Fraenkel and Noether, who still dealt mostly with commutative rings.



However, it soon became apparent that there were too many instances where commutativity of multiplication did not hold. You had famously the quaternions, of course, but you also had matrices and, more generally, the endomorphism ring of an abelian group (where “multiplication” is composition of functions). So that we have two different, related, notions: commutative rings and noncommutative rings, just like we have noncommutative groups and commutative/abelian groups.



Now, why do this with multiplication and not with addition? Well, if you take your definition of ring above, which includes a unity, but you drop condition (3) (that is, you require everything except you do not require that addition be commutative), it turns out that the other eight axioms force commutativity of addition.



Indeed, suppose you have a structure $(R,+,cdot,0,1)$ that satisfies axioms (1), (2), and (4)-(9) above. I claim that one can deduce (3). Indeed, let $a,bin R$. Then using associativity on the left first, and associativity on the right second, we have
$$begin{align*}
(1+1)(a+b) &= 1(a+b) + 1(a+b) = a+b+a+b\
(1+1)(a+b) &= (1+1)a + (1+1)b = a+a+b+b.
end{align*}$$

From this we get that $a+b+a+b = a+a+b+b$. Now add the inverse of $a$ on the left and the inverse of $b$ on the left on both sides to get
$$begin{align*}
(-a) + a + b + a + b + (-b) &= 0+b+a+0 = b+a\
(-a) + a + a + b + b + (-b) &= 0+a+b+0 = a+b
end{align*}$$

Thus, we conclude that $a+b=b+a$. That is, commutativity of addition is a consequence of the other eight axioms.



The reason we include it is two-fold: one, is that it is much nicer to say that the first few axioms force $(R,+)$ to be a commutative/abelian group. The second is that it is also common to consider rings without unity, and if we do that, then it is no longer true that addition is forced to be commutative. To see this, note that if $(G,cdot)$ is any group with identity element $e_G$, and we define a multiplication on $G$ by letting $a*b=e_G$ for all $a,bin G$, then $(G,cdot,*)$ satisfies axioms (1)-(8) given above. But if the original group is not commutative, then the “addition” in this ring is not commutative. So if we want to consider rings without unity, we do want to explicitly require addition to be commutative.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
    $endgroup$
    – Bill Dubuque
    1 hour ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3134219%2fwhy-is-commutativity-optional-in-multiplication-for-rings%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

The first rings that were considered were generally commutative; polynomial rings, then from work of Dedekind in number fields. The properties were then abstracted Fraenkel and Noether, who still dealt mostly with commutative rings.



However, it soon became apparent that there were too many instances where commutativity of multiplication did not hold. You had famously the quaternions, of course, but you also had matrices and, more generally, the endomorphism ring of an abelian group (where “multiplication” is composition of functions). So that we have two different, related, notions: commutative rings and noncommutative rings, just like we have noncommutative groups and commutative/abelian groups.



Now, why do this with multiplication and not with addition? Well, if you take your definition of ring above, which includes a unity, but you drop condition (3) (that is, you require everything except you do not require that addition be commutative), it turns out that the other eight axioms force commutativity of addition.



Indeed, suppose you have a structure $(R,+,cdot,0,1)$ that satisfies axioms (1), (2), and (4)-(9) above. I claim that one can deduce (3). Indeed, let $a,bin R$. Then using associativity on the left first, and associativity on the right second, we have
$$begin{align*}
(1+1)(a+b) &= 1(a+b) + 1(a+b) = a+b+a+b\
(1+1)(a+b) &= (1+1)a + (1+1)b = a+a+b+b.
end{align*}$$

From this we get that $a+b+a+b = a+a+b+b$. Now add the inverse of $a$ on the left and the inverse of $b$ on the left on both sides to get
$$begin{align*}
(-a) + a + b + a + b + (-b) &= 0+b+a+0 = b+a\
(-a) + a + a + b + b + (-b) &= 0+a+b+0 = a+b
end{align*}$$

Thus, we conclude that $a+b=b+a$. That is, commutativity of addition is a consequence of the other eight axioms.



The reason we include it is two-fold: one, is that it is much nicer to say that the first few axioms force $(R,+)$ to be a commutative/abelian group. The second is that it is also common to consider rings without unity, and if we do that, then it is no longer true that addition is forced to be commutative. To see this, note that if $(G,cdot)$ is any group with identity element $e_G$, and we define a multiplication on $G$ by letting $a*b=e_G$ for all $a,bin G$, then $(G,cdot,*)$ satisfies axioms (1)-(8) given above. But if the original group is not commutative, then the “addition” in this ring is not commutative. So if we want to consider rings without unity, we do want to explicitly require addition to be commutative.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
    $endgroup$
    – Bill Dubuque
    1 hour ago
















5












$begingroup$

The first rings that were considered were generally commutative; polynomial rings, then from work of Dedekind in number fields. The properties were then abstracted Fraenkel and Noether, who still dealt mostly with commutative rings.



However, it soon became apparent that there were too many instances where commutativity of multiplication did not hold. You had famously the quaternions, of course, but you also had matrices and, more generally, the endomorphism ring of an abelian group (where “multiplication” is composition of functions). So that we have two different, related, notions: commutative rings and noncommutative rings, just like we have noncommutative groups and commutative/abelian groups.



Now, why do this with multiplication and not with addition? Well, if you take your definition of ring above, which includes a unity, but you drop condition (3) (that is, you require everything except you do not require that addition be commutative), it turns out that the other eight axioms force commutativity of addition.



Indeed, suppose you have a structure $(R,+,cdot,0,1)$ that satisfies axioms (1), (2), and (4)-(9) above. I claim that one can deduce (3). Indeed, let $a,bin R$. Then using associativity on the left first, and associativity on the right second, we have
$$begin{align*}
(1+1)(a+b) &= 1(a+b) + 1(a+b) = a+b+a+b\
(1+1)(a+b) &= (1+1)a + (1+1)b = a+a+b+b.
end{align*}$$

From this we get that $a+b+a+b = a+a+b+b$. Now add the inverse of $a$ on the left and the inverse of $b$ on the left on both sides to get
$$begin{align*}
(-a) + a + b + a + b + (-b) &= 0+b+a+0 = b+a\
(-a) + a + a + b + b + (-b) &= 0+a+b+0 = a+b
end{align*}$$

Thus, we conclude that $a+b=b+a$. That is, commutativity of addition is a consequence of the other eight axioms.



The reason we include it is two-fold: one, is that it is much nicer to say that the first few axioms force $(R,+)$ to be a commutative/abelian group. The second is that it is also common to consider rings without unity, and if we do that, then it is no longer true that addition is forced to be commutative. To see this, note that if $(G,cdot)$ is any group with identity element $e_G$, and we define a multiplication on $G$ by letting $a*b=e_G$ for all $a,bin G$, then $(G,cdot,*)$ satisfies axioms (1)-(8) given above. But if the original group is not commutative, then the “addition” in this ring is not commutative. So if we want to consider rings without unity, we do want to explicitly require addition to be commutative.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
    $endgroup$
    – Bill Dubuque
    1 hour ago














5












5








5





$begingroup$

The first rings that were considered were generally commutative; polynomial rings, then from work of Dedekind in number fields. The properties were then abstracted Fraenkel and Noether, who still dealt mostly with commutative rings.



However, it soon became apparent that there were too many instances where commutativity of multiplication did not hold. You had famously the quaternions, of course, but you also had matrices and, more generally, the endomorphism ring of an abelian group (where “multiplication” is composition of functions). So that we have two different, related, notions: commutative rings and noncommutative rings, just like we have noncommutative groups and commutative/abelian groups.



Now, why do this with multiplication and not with addition? Well, if you take your definition of ring above, which includes a unity, but you drop condition (3) (that is, you require everything except you do not require that addition be commutative), it turns out that the other eight axioms force commutativity of addition.



Indeed, suppose you have a structure $(R,+,cdot,0,1)$ that satisfies axioms (1), (2), and (4)-(9) above. I claim that one can deduce (3). Indeed, let $a,bin R$. Then using associativity on the left first, and associativity on the right second, we have
$$begin{align*}
(1+1)(a+b) &= 1(a+b) + 1(a+b) = a+b+a+b\
(1+1)(a+b) &= (1+1)a + (1+1)b = a+a+b+b.
end{align*}$$

From this we get that $a+b+a+b = a+a+b+b$. Now add the inverse of $a$ on the left and the inverse of $b$ on the left on both sides to get
$$begin{align*}
(-a) + a + b + a + b + (-b) &= 0+b+a+0 = b+a\
(-a) + a + a + b + b + (-b) &= 0+a+b+0 = a+b
end{align*}$$

Thus, we conclude that $a+b=b+a$. That is, commutativity of addition is a consequence of the other eight axioms.



The reason we include it is two-fold: one, is that it is much nicer to say that the first few axioms force $(R,+)$ to be a commutative/abelian group. The second is that it is also common to consider rings without unity, and if we do that, then it is no longer true that addition is forced to be commutative. To see this, note that if $(G,cdot)$ is any group with identity element $e_G$, and we define a multiplication on $G$ by letting $a*b=e_G$ for all $a,bin G$, then $(G,cdot,*)$ satisfies axioms (1)-(8) given above. But if the original group is not commutative, then the “addition” in this ring is not commutative. So if we want to consider rings without unity, we do want to explicitly require addition to be commutative.






share|cite|improve this answer











$endgroup$



The first rings that were considered were generally commutative; polynomial rings, then from work of Dedekind in number fields. The properties were then abstracted Fraenkel and Noether, who still dealt mostly with commutative rings.



However, it soon became apparent that there were too many instances where commutativity of multiplication did not hold. You had famously the quaternions, of course, but you also had matrices and, more generally, the endomorphism ring of an abelian group (where “multiplication” is composition of functions). So that we have two different, related, notions: commutative rings and noncommutative rings, just like we have noncommutative groups and commutative/abelian groups.



Now, why do this with multiplication and not with addition? Well, if you take your definition of ring above, which includes a unity, but you drop condition (3) (that is, you require everything except you do not require that addition be commutative), it turns out that the other eight axioms force commutativity of addition.



Indeed, suppose you have a structure $(R,+,cdot,0,1)$ that satisfies axioms (1), (2), and (4)-(9) above. I claim that one can deduce (3). Indeed, let $a,bin R$. Then using associativity on the left first, and associativity on the right second, we have
$$begin{align*}
(1+1)(a+b) &= 1(a+b) + 1(a+b) = a+b+a+b\
(1+1)(a+b) &= (1+1)a + (1+1)b = a+a+b+b.
end{align*}$$

From this we get that $a+b+a+b = a+a+b+b$. Now add the inverse of $a$ on the left and the inverse of $b$ on the left on both sides to get
$$begin{align*}
(-a) + a + b + a + b + (-b) &= 0+b+a+0 = b+a\
(-a) + a + a + b + b + (-b) &= 0+a+b+0 = a+b
end{align*}$$

Thus, we conclude that $a+b=b+a$. That is, commutativity of addition is a consequence of the other eight axioms.



The reason we include it is two-fold: one, is that it is much nicer to say that the first few axioms force $(R,+)$ to be a commutative/abelian group. The second is that it is also common to consider rings without unity, and if we do that, then it is no longer true that addition is forced to be commutative. To see this, note that if $(G,cdot)$ is any group with identity element $e_G$, and we define a multiplication on $G$ by letting $a*b=e_G$ for all $a,bin G$, then $(G,cdot,*)$ satisfies axioms (1)-(8) given above. But if the original group is not commutative, then the “addition” in this ring is not commutative. So if we want to consider rings without unity, we do want to explicitly require addition to be commutative.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 1 hour ago

























answered 2 hours ago









Arturo MagidinArturo Magidin

264k34588915




264k34588915












  • $begingroup$
    There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
    $endgroup$
    – Bill Dubuque
    1 hour ago


















  • $begingroup$
    There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
    $endgroup$
    – Bill Dubuque
    1 hour ago
















$begingroup$
There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
$endgroup$
– Bill Dubuque
1 hour ago




$begingroup$
There is an entire thread on the 4th paragragh, see Why is ring addition commuattive?
$endgroup$
– Bill Dubuque
1 hour ago


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3134219%2fwhy-is-commutativity-optional-in-multiplication-for-rings%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

“%fieldName is a required field.”, in Magento2 REST API Call for GET Method Type The Next...

How to change City field to a dropdown in Checkout step Magento 2Magento 2 : How to change UI field(s)...

變成蝙蝠會怎樣? 參考資料 外部連結 导航菜单Thomas Nagel, "What is it like to be a...