Strongly Idempotent Seminearrings and Their Prime Ideal Spaces

Abstract

A right seminearring is a set R together with two binary operations + and · such that (R, +) and (R, ·) are semigroups and for all a, b, c ∈ R: (a + b)c = ac + bc ([6, 9, 11]). A natural example of a right seminearring is the set M(S) of all mappings on an additively written semigroup S with pointwise addition and composition. A right seminearring R is said to have an absorbing zero 0 if a + 0 = 0 + a = a and a · 0 = 0 · a = 0 hold for all a ∈ R. A seminearring homomorphism between seminearrings R and R’ is a map Φ : R → R’ satisfying Φ(a + b) = Φ(a) + Φ(b) and Φ(ab) = Φ(a) Φ(b) for all a, b ∈ R. An ideal of a seminearring with an absorbing zero is the kernel of a seminearring homomorphism ([7]). Generalizing this definition, we call a non-empty subset I of a seminearring R a right (left) S-ideal if (i) for all x, y ∈ I, x + y ∈ I, and (ii) for all x ∈ I and r ∈ R, xr (rx) ∈ I. The word S-ideal will always mean a subset of R which is both a left and a right S-ideal. If R is moreover a unitary nearring then S-ideals of R are just the R-subgroups of R (see [9, p. 14]). If R is a unitary ring then S-ideals are the ideals of R in the usual ring-theoretical sense. However, if R is not a unitary ring (that is, R is a ring without an identity element) then S-ideals of R need not be ideals in the usual sense. Let R be the subring xZ[x] of the polynomial ring Z[x] over the ring of integers Z. Then (R, +, ·) is a commutative ring without identity.