Nonsingular rings with essential socles

G. Ivanov
1974 Journal of the Australian Mathematical Society  
Communicated by M. F. Newman This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2],
more » ... y and Rutter [2], and for the semiprimary case by Zaks [12] . Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1. The results of section 2 are used in section 3 to investigate the structure of nonsingular QF-3 rings with finite identities. Semi-primary QF-3 rings which are also partially PP rings were first studied by Harada [8], [9] who showed that each of these rings has a semi-simple Artinian ring as a left, and a right, injective hull and that a semi-primary hereditary QF-3 ring is generalized uniserial. Colby and Rutter [3] proved these results for the semi-perfect case and later [4] showed that an arbitrary nonsingular left and right QF-3 ring has a semi-simple Artinian ring as a left, and a right, essential extension. In section 3 a matrix representation of nonsingular QF-3 rings with finite identities is used to determine the structure of left and right QF-3 rings and of semi-perfect QF-3 rings with various abundances of projective ideals. The following conventions, terminology and notation are used throughout the paper. All exceptions to these are specifically mentioned so there are, hopefully, no ambiguities. All rings have identities and all modules are unital. The adjective 358 use, available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s1446788700017110 fatcat:faikjof7nrhkffvxuzirc6v24a