Associative rings and algebras Definition

Stably finite ring

Builds on Dedekind-finite ring Matrix ring

Definition

Let $R$ be a ring with $1$ . For $n \ge 1$ , let $M_n(R)$ be the matrix ring of $n \times n$ matrices over $R$ . The ring $R$ is stably finite if $M_n(R)$ is Dedekind-finite for every $n \ge 1$ . That is, for all $n \ge 1$ and $A, B \in M_n(R)$ , the equation $AB = I_n$ implies $BA = I_n$ .

Remarks

Example: every commutative ring $R$ is stably finite. If $AB = I_n$ in $M_n(R)$ , then $\det(A) \det(B) = 1$ , so $A$ has the inverse $\det(A)^{-1} \operatorname{adj}(A)$ , and $BA = A^{-1}(AB)A = I_n$ .

Non-example: the ring $\End(V)$ of linear maps on an infinite-dimensional vector space $V$ is not stably finite, since already $M_1(\End(V)) = \End(V)$ contains $a, b$ with $ab = 1 \ne ba$ .

Taking $n = 1$ shows that a stably finite ring is Dedekind-finite. The converse fails: there are Dedekind-finite rings $R$ for which $M_2(R)$ is not Dedekind-finite. Equivalently, $R$ is stably finite if for every $n \ge 1$ the left module $R^n$ is not isomorphic to $R^n \oplus N$ for a nonzero module $N$ . A nonzero stably finite ring has invariant basis number.

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Tag	`JGVDA1N106`
URI	`m2://JGVDA1N106`
Kind	definition
Writing	W0 stub
MSC	16U99, 16D40
Defines	Stably finite ring
Uses	Dedekind-finite ring, Matrix ring

Stably finite ring

Definition

Remarks

Reading path

Further reading

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