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I know that if a ring has a multiplicative identity, then the multiplicative identity must be unique. Are there simple-to-describe examples of rings with two (or more) multiplicative right-identities?

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Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = aa = a, ba = bb = b \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the first letter in it. The resulting semigroup ring is literally the free ring on two not necessarily identical right identities.

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Not that it matters, but doesn't your example have two left identities, not two right identities? – Richard Stanley May 1 '11 at 15:38
Whoops. Thanks for the correction. – Qiaochu Yuan May 1 '11 at 16:15

I was browsing some "What is...?" threads, and bumped in here: Let $n$ be a positive integer, and consider the subring of the opposite of the ring of $n$-by-$n$ matrices over a commutative unital ring $\mathbb A = (A, +, \cdot)$ consisting of those matrices all of whose rows, except at most for the first, are zero; this has $|A|^{n-1}$ right identities, given by those matrices whose first row is any vector of $A^n$ with first element equal to $1_\mathbb{A}$.

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