An isomorphism problem is considered for generalized matrix rings with values in a given ring R. An exhaustive answer is given for the case of a commutative domain R and a commutative local ring R.For generalized (or formal) matrix rings and their connections with Morita contexts, we ask the reader to consult [1(Chap. 12)-3]. These rings have been intensively studied as of late (see, e.g., [4][5][6][7][8]). Generalized matrix rings appear in the theory of rings and modules in quite various situations. Such rings, for instance, have found interesting application in treating endomorphism rings of Abelian groups [9][10][11]. Triangular generalized matrix rings, as is well known, are a source for constructing rings with asymmetric properties. A chapter in [2] is concerned with just these rings.A generalized matrix ring is defined by two bimodule homomorphisms. The choice of another pair of homomorphisms generally leads up to another ring. There arises a problem of classifying generalized matrix rings in relation to corresponding pairs of bimodule homomorphisms. In the present paper we solve this problem for generalized matrix rings with values in a given ring R. The answer is obtained for the case of a commutative domain R and a commutative local ring R.We give the definition of a generalized matrix ring. Let R and S be rings, M an R-S-bimodule, and N an S-R-bimodule. Assume that there are bimodule homomorphisms ϕ : M ⊗ S N → R and ψ : N ⊗ R M → S satisfying the following associativity conditions: (mn)m = m(nm ) and (nm)n = n(mn ) for all m, m ∈ M and n, n ∈ N . Put mn = ϕ(m ⊗ n) and nm = ψ(n ⊗ m). Consider a set R M N S of all matrices r m n s , where r ∈ R, s ∈ S, m ∈ M , and n ∈ N . With respect to operations specified as in the usual matrix ring, this set is a ring, which is called a generalized matrix ring (of order 2). We fix the letter K to denote the introduced ring R M N S . If emphasis should be laid on the fact that K is constructed using homomorphisms ϕ and ψ, then we write K(ϕ, ψ). A ring K(ϕ, ψ) with zero homomorphisms ϕ and ψ is said to be trivial. Triangular generalized matrix rings pertain to trivial ones (if M = 0 or N = 0). In a similar way, we can define a generalized matrix ring of any order n > 2. Such a ring is naturally isomorphic to a generalized matrix ring of order 2 (the isomorphism is obtained by partitioning every matrix into four blocks). Usually, from technical considerations, order 2 generalized matrix rings are considered. We do so in the present paper, too. A class of generalized matrix rings coincides with a class pr. Mira 3-60, Tomsk, 634057 Russia; krylov@math.tsu.ru.
