Weakly primitive rings

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Everyone who has aquainted himself with some ring theory is familiar with Jacobson's classical structure theorem for primitive rings. This thesis is based on the study of a class of prime rings which satisfy a weaker criterion than the class of primitive rings. The study of prime rings plays a significant role in modern ring theory, and since Jacobson's result includes the Wedderburn-Artin Theorem, which is the major structure theorem for rings possessing descending chain conditions, as a special case, the importance of Jacobson's result cannot be understated. The central concepts and results presented here were originally proved by Zelmanowitz in [9] and Amitsur in [1]. We call a module compressible provided that for any non- zero submodule N of M, HomR(MN) contains a monomorphism, and is called monoform when each non- zero element of HomR(N,M) is injective. Our major result (Zelmanowitz [9]) shows that a ring possessing a faithful compressible monoform module satisfies a weakened density criterion (such rings are called weakly primitive). Next we define a module R Mto be prime [ semiprime] if for any non- zero elements m,m 1 E M there exists f E HomR(M,R) such that (nif)m 1 0 0 [(mnf)m 0 0] and we specialize Zelmanowitz' result to the case of prime and semiprime modules with certain maximum conditions ( Zelmanowitz [ 8 ]). Finally, we give an exposition of some results by Nicholson, Watters and Zelmanowitz in [7] concerning derived rings of weakly primitive rings. Among other interesting results we prove that polynomial rings and certain group rings over weakly primitive rings are weakly primitive as well.
Bibliography: p. 84.
Bryden, J. M. (1979). Weakly primitive rings (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/18215