Extensions of a ring by a ring with a bimodule structure

A type of ring extension is considered that was introduced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings. This paper is a sequel to [2 ], in which a class of examples was given of a type of ring extension defined in terms of two functions. Here we exhibit some other functions that may be used to construct such extensions, and show that in certain cases (in particular, when the first ring is an integral domain and the second is a commutative ring with identity), the functions must have a prescribed form. We also characterize this type of ring extension, which Szendrei defined directly by the ring operations, in terms of the manner in which the two given rings are embedded in the extension. The reader is referred to [2] for background. Only the basic definition is repeated here. Let A and B be rings. We define the ring A*B to be the direct sum of A and B as additive groups, with multiplication given by (1) (a, b)(c, d) = (ac + {b, d}, ad + bo-c + bd), where ois a homomorphism from A onto a ring of permutable bimultiplications of B, and {I , *} is a biadditive function from B XB into A satisfying the equations (2) bo-1c, d} = 1b, c)d, (3) 1{b, cd} = {bc, di, (4) {b ?oaC} = {boa, c}, (5) aOabC}c = a{b,c}, and {b,co-a} = {b,c}a, for all aCA and all b, c, deB. As noted in [2], when B has an identity, (4) is redundant. The inverse of a homomorphism 4 will be denoted by O', even when 4 is not a monomorphism. Received by the editors September 2, 1969. AMS Subject Classifications. Primary 1680, 1380; Secondary 1315.