Projections of Semilocal Rings

Abstract

Associative rings are considered. By a lattice isomorphism (or projection) of a ring R onto a ring R^φ we mean an isomorphism φ of the subring lattice L(R) of a ring R onto the subring lattice L(R^φ) of a ring R^φ. Let M_n(GF(p^k)) be the ring of all square matrices of order n over a finite field GF(p^k), where n and k are natural numbers, p is a prime. A finite ring R with identity is called a semilocal (primary) ring if R/RadR ≅ M_n(GF(p^k)). It is known that a finite ring R with identity is a semilocal ring iff R ≅ M_n(K) and K is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if φ is a projection of a ring R = M_n(K), where K is an arbitrary finite local ring, onto a ring R^φ, then R^φ = Mn(K′), in which case K′ is a local ring lattice-isomorphic to the ring K. We thus prove that the class of semilocal rings is lattice definable.