Regular modules

Author:
J. Zelmanowitz

Journal:
Trans. Amer. Math. Soc. **163** (1972), 341-355

MSC:
Primary 16.56

MathSciNet review:
0286843

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Abstract: In analogy to the elementwise definition of von Neumann regular rings an -module is called regular if given any element there exists with . Other equivalent definitions are possible, and the basic properties of regular modules are developed. These are applied to yield several characterizations of regular self-injective rings. The endomorphism ring of a regular module is examined. It is in general a semiprime ring with a regular center. An immediate consequence of this is the recently observed fact that the endomorphism ring of an ideal of a commutative regular ring is again a commutative regular ring. Certain distinguished subrings of are also studied. For example, the ideal of consisting of the endomorphisms with finite-dimensional range is a regular ring, and is simple when the socle of is homogeneous. Finally, the self-injectivity of is shown to depend on the quasi-injectivity of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0286843-3

Keywords:
von Neumann regular ring,
regular module,
finite-dimensional module,
projective module,
flat module,
endomorphism ring,
endomorphisms with finite-dimensional range,
quasi-injective module

Article copyright:
© Copyright 1972
American Mathematical Society