Vol. 59, No. 2, 1975

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Some aspects of T-nilpotence. II: Lifting properties over T-nilpotent ideals

Barry J. Gardner

Vol. 59 (1975), No. 2, 445–453

It has been shown by Nǎstǎsescu and Popescu that every nonzero (left, unital) module over a ring R has a simple submodule if and only if the Jacobson radical J of R is right T-nilpotent and every nonzero R∕J-module has a simple submodule. The work presented here arose largely from an attempt to find a general framework for results like this.

In §2 it is shown that if R has a right T-nilpotent ideal I, then a bijection from the torsion classes of R∕I-modules to those of R-Modules can be obtained by associating with each 𝒯 Mod(R∕I) the lower radical class it defines as a class of R-modules. §3 contains applications involving the lifting of torsion properties and in §4 it is shown that if R has a right T-nilpotent ideal I such that R∕I is the direct sum of its torsion and divisible ideals, then R has this property also.

Mathematical Subject Classification
Primary: 16A22
Received: 19 March 1975
Published: 1 August 1975
Barry J. Gardner