Do commutative rings without unity have the IBN property?

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Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.




Does $R$ still have the Invariant Basis Number (IBN) property?




Recall that a ring is said to have the IBN property if $R^m cong R^n Rightarrow m=n$.



All commutative rings have the IBN property, but the standard proof I know makes an essential use of the existence of a maximal ideal by Zorn's Lemma and passing to the residue field, which depends on the presence of a unit.










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  • 1




    Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
    – LSpice
    2 hours ago







  • 5




    The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
    – Keith Kearnes
    2 hours ago






  • 3




    It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
    – Benjamin Steinberg
    1 hour ago














up vote
1
down vote

favorite












Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.




Does $R$ still have the Invariant Basis Number (IBN) property?




Recall that a ring is said to have the IBN property if $R^m cong R^n Rightarrow m=n$.



All commutative rings have the IBN property, but the standard proof I know makes an essential use of the existence of a maximal ideal by Zorn's Lemma and passing to the residue field, which depends on the presence of a unit.










share|cite|improve this question



















  • 1




    Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
    – LSpice
    2 hours ago







  • 5




    The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
    – Keith Kearnes
    2 hours ago






  • 3




    It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
    – Benjamin Steinberg
    1 hour ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.




Does $R$ still have the Invariant Basis Number (IBN) property?




Recall that a ring is said to have the IBN property if $R^m cong R^n Rightarrow m=n$.



All commutative rings have the IBN property, but the standard proof I know makes an essential use of the existence of a maximal ideal by Zorn's Lemma and passing to the residue field, which depends on the presence of a unit.










share|cite|improve this question















Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.




Does $R$ still have the Invariant Basis Number (IBN) property?




Recall that a ring is said to have the IBN property if $R^m cong R^n Rightarrow m=n$.



All commutative rings have the IBN property, but the standard proof I know makes an essential use of the existence of a maximal ideal by Zorn's Lemma and passing to the residue field, which depends on the presence of a unit.







ac.commutative-algebra






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share|cite|improve this question













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edited 2 hours ago

























asked 2 hours ago









M.G.

2,67522537




2,67522537







  • 1




    Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
    – LSpice
    2 hours ago







  • 5




    The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
    – Keith Kearnes
    2 hours ago






  • 3




    It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
    – Benjamin Steinberg
    1 hour ago












  • 1




    Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
    – LSpice
    2 hours ago







  • 5




    The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
    – Keith Kearnes
    2 hours ago






  • 3




    It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
    – Benjamin Steinberg
    1 hour ago







1




1




Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
– LSpice
2 hours ago





Simply adjoint a unit to $R$ to obtain $R_1$ (e.g., $R_1 = R[x]/(r x - r : r in R)$). Then an isomorphism $R^m cong R^n$ automatically extends to an isomorphism $R_1^m cong R_1^n$, and you can use the IBN property for $R_1$.
– LSpice
2 hours ago





5




5




The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
– Keith Kearnes
2 hours ago




The answer is "not necessarily". For a counterexample, let $R = oplus^omega mathbb Z$ with zero multiplication.
– Keith Kearnes
2 hours ago




3




3




It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
– Benjamin Steinberg
1 hour ago




It should be pointed out that $R^n$ doesn't have a basis either when $R$ is non-unital.
– Benjamin Steinberg
1 hour ago










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(I will write my comment as an answer.)



The answer is "not necessarily". For a counterexample, let $R=oplus^omega mathbb Z$
with zero multiplication.



However let me add this: when $R$ is nonunital, $R^n$ is not the $n$-generated free $R$-module.






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    1 Answer
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    up vote
    4
    down vote













    (I will write my comment as an answer.)



    The answer is "not necessarily". For a counterexample, let $R=oplus^omega mathbb Z$
    with zero multiplication.



    However let me add this: when $R$ is nonunital, $R^n$ is not the $n$-generated free $R$-module.






    share|cite|improve this answer


























      up vote
      4
      down vote













      (I will write my comment as an answer.)



      The answer is "not necessarily". For a counterexample, let $R=oplus^omega mathbb Z$
      with zero multiplication.



      However let me add this: when $R$ is nonunital, $R^n$ is not the $n$-generated free $R$-module.






      share|cite|improve this answer
























        up vote
        4
        down vote










        up vote
        4
        down vote









        (I will write my comment as an answer.)



        The answer is "not necessarily". For a counterexample, let $R=oplus^omega mathbb Z$
        with zero multiplication.



        However let me add this: when $R$ is nonunital, $R^n$ is not the $n$-generated free $R$-module.






        share|cite|improve this answer














        (I will write my comment as an answer.)



        The answer is "not necessarily". For a counterexample, let $R=oplus^omega mathbb Z$
        with zero multiplication.



        However let me add this: when $R$ is nonunital, $R^n$ is not the $n$-generated free $R$-module.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 20 mins ago

























        answered 26 mins ago









        Keith Kearnes

        5,65412742




        5,65412742



























             

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