A ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism/ from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs. Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this

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... r (finite) special principal ideal rings that are absolute subretracts are studied. All rings in this paper are associative and commutative, but do not necessarily contain an identity. For a ring R we denote by Var(/£) the variety generated by R (cf. [5]). Recall (cf. [4]) that a ring R with identity is called a special principal ideal ring if R is a local principal ideal ring with (nonzero) nilpotent maximal ideal M. Obviously M -(3 (/?), the prime radical of R. In [3] several notions of injectiveness within a variety of rings are studied. Particular attention is given to absolute subretracts. A ring R is said to be an absolute subretract if for every ring S in Var(#) and for every ring monomorphism/: R-+ S there exists a ring morphism g: S -• R such that gf is the identity mapping. Or equivalently, for every such morphism/ there exists a two-sided ideal M of S such that S = f(R) 0 M, a direct sum as/(#)-modules. Gardner and Stewart in [3] characterize directly indecomposable absolute subretracts R with R 2 = 0. However very little is known for non-semiprime rings with R 2 ^ 0. Actually only one example of a ring R of this kind is included in [3], namely R -Z2 [X]/ (X 2 ). Clearly R is a finite special principal ideal ring. The aim of this paper is to give necessary and sufficient conditions for a finite special principal ideal ring R to be an absolute subretract. In general we obtain necessary conditions; but, if the characteristic of R, char(/?), is not a power of 2, these conditions turn out to be sufficient too. This result gives us more examples of non-semiprime absolute subretracts with identity. PROPOSITION 1. Let Rbea special principal ideal ring. IfR is an absolute subretract, then (3 (R) 3 = 0. If, moreover, char(/?/ (3 (/?)) ± 2, thenf3(R) 2 = 0.