A finite dimensional commutative algebra with

A finite dimensional commutative algebra with A finite dimensional commutative algebra with nilpotent radical, an identity modulo the radical, but no global identity.

******************************

Let A = C² with multiplication (a,b)(c,d) = (ac,0) (a,b,c,d Î C). Clearly A² = A. Its radical is R = {(0,b);b Î C} and [ A/ R] @ C. The identity of [ A/ R] lifts to the idempotent (1,0) in A [Ric, Theorem 2.3.9], but there is no identity in A.

Ref.

[Ric] C.E. Rickart, General theory of Banach algebras, Princeton, Van Nastrand, 1960.

File translated from T_EX by T_TH, version 2.70.
On 22 Feb 2001, 00:15.