Abstract
A commutative ring R with identity is called indecomposable if R has only the trivial idempotents, i.e. in R holds: V Vo(V 2 = Vo---~ v0 = 0 v v0 = 1). Therefore the theory of all commutative indecomposable rings with identity is axiomatizable by a finite set of universal sentences. A commutative indecomposable ring with identity is said to be an algebraically closed indecomposable ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative indecomposabte extension of R already has a solution in R. We will show in the present paper results for commutative indecomposable rings with identity similar to those proved by Cherlin for commutative rings (2) and by us (see [7]) for commutative local rings. Our main results are the following:
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