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How do one usually define the general linear group over a ring $R$, denoted by $\text{GL}(n,R)$. I was told in a paper that $\text{GL}(n,R)$ is a group, and I presumed that $$\text{GL}(n,R)=\{A\in M_{n\times n}(R)|\text{det}(A)~\mbox{is a unit in}~R\}.$$However, I tried google it and found $$\text{GL}(n,R)=\{A\in M_{n\times n}(R)|\text{det}(A)\neq0\}.$$See for example, . As $R$ is not necessarily a unital ring, so it would happen that $\text{GL}(n,R)$ is not a group. Could any expert tell me which understanding is correct? And also, could you recommend any textbook which provides detailed discussion about this kind of group? I need to learn this more, thank you very much!

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1 Answer

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If $R$ is a commutative ring with identity, then for an integer $n\geq 1$, $\mathrm{GL}_n(R)$ is the set of $n\times n$ matrices in $g\in\mathrm{M}_n(R)$ (coefficients in $R$) such that $\mathrm{det}(g)\in R^\times$. This is precisely the group of invertible elements of the ring $\mathrm{M}_n(R)$ of matrices.

If $R$ is a field, then $R^\times=R\setminus\{0\}$, so in this case, the condition $\det(g)\in R^\times$ is equivalent to $\det(g)\neq 0$, but this is not true for (non-zero) commutative rings which aren't fields.

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