ringsentrumiks

Ringsentrumiks are a class of algebraic structures that generalize rings by incorporating a distinguished central subring, called the centrum. A ringsentrumik consists of a ring R together with a subring Z contained in the center of R, denoted Center(R). The centrum acts on R in a way that makes R into a Z-algebra, with the multiplication in R being Z-bilinear: for z in Z and r, s in R, z(r s) = (z r) s = r(z s). Thus Z controls a scalar-like interaction with the elements of R while remaining central.

In this framework, a morphism of ringsentrumiks from (R, Z) to (R', Z') is a ring homomorphism f: R -> R' that maps Z into Z' in a way compatible with the inclusion into Center(R'). Equivalently, it preserves both the ring structure of R and the central actions defined by the centrum subrings.

Key concepts include ringsentrumik ideals, which are two-sided ideals of R stable under the action of Z, and ringsentrumik-simple rings, which have no nontrivial Z-stable two-sided ideals. The presence of Z often guides decomposition and representation-theoretic considerations, allowing one to study R through its interaction with the central subring.

Examples illustrate the standard case: if k is a field and R is a k-algebra with Z identified with k via scalar matrices, then R becomes a ringsentrumik with centrum Z = k. The classic instance is the full matrix ring R = M_n(k), where Z corresponds to the scalar matrices, a central subring that governs the Z-algebra structure.

Ringsentrumiks appear in discussions of central actions on algebras and in explorations of how central substructures influence decomposition, representation theory, and aspects of noncommutative geometry. See also ring, center, centroid, and noncommutative algebra.