Vol. 45, No. 2, 1973

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Essential products of nonsingular rings

Kenneth R. Goodearl

Vol. 45 (1973), No. 2, 493–505

By an essential product of two rings is meant a subdirect product which contains an essential right ideal of the direct product. The aim of this paper is to investigate the utility of this concept in the study of nonsingular rings. The first section derives some basic properties of essential products and develops some criteria for recognizing essential products. In the second section, a study of the socles of nonsingular modules leads to a theorem that any nonsingular ring is an essential product of a ring with essential socle and a ring with zero socle. The third section is devoted to a theorem which tells when an essential product can be a splitting ring, i.e., a ring such that the singular submodule of any right module is a direct summand. In the final section, this theorem is used to construct two examples of splitting rings of types previously unknown.

Mathematical Subject Classification
Primary: 16A48
Received: 25 February 1972
Published: 1 April 1973
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States