Weakly primitive rings
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  and Amitsur in . 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 ) 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  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 Weakly primitive rings (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/18215