Semi-Primary QF-3 Rings

Abstract

A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition. R is called a left QF-3 ring if it contains a faithful projective injective left ideal.If R is semi-primary and left QF-3> then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left i?-module In this note we determine the class of semi-primary rings in which the notions of QF-3 and QF-3+ coincide.Next, we show that the characterization of QF-3 + rings given by Wu, Mochizuki, and Jans [7] for rings with the property that direct products of projective modules are projective, can be used to characterize semi-primary QF-3 rings. Finally, we give some results relating the notions of torsionless and torsion-free modules as defined by H. Bass [1] and A.W. Goldie [3]. In particular we show that if R is semi-primary, these notions coincide if and only if R is left QF-3 and has zero left singular ideal.

On left QF− 3 rings

“…For the proofs we shall refer to that of Proposition 2 and Theorem 2 in [4], which are known to be valid, if we consider that for right perfect rings, every nonzero left module has nonzero socle and for left perfect rings, every projective module is isomorphic to a direct sum of primitive left ideals. Proof.…”

“…Our main purpose in this note is to introduce some generalizations of results for QF-3 algebras [11], [12], [15], [16], [17] and semi-primary QF-3 rings [4], [6], [13], [14] to the above generalized classes of rings.…”

“…Recently, the characterization of Artinian QF-3 rings due to Wu, Mochizuki and Jans [19] suggested the notion of QF-3' rings to Colby and Rutter [4] and Kato [7]. In [4] it was proved that semi-primary 1) left QF-3' rings are not necessarily left QF-3+, however "left QF-3'" implies "left QF-3" for semi-primary rings.…”

“…In [4] it was proved that semi-primary 1) left QF-3' rings are not necessarily left QF-3+, however "left QF-3'" implies "left QF-3" for semi-primary rings. Without the proof we shall state in §3 that the same result holds for perfect rings, since the proof in [4] is available for this case. Moreover, we shall prove by duality of modules and the result proved in the first part of §2 the notions of two sided QF-3', QF-3+ and QF-3 are identical for perfect rings.…”

A characterization of QF− 3 rings

“…In this case E( R R) need not be protective and R need not be right QF-Z (cf. [3] and [8]). However, a perfect ring is QF-Z if and only if both E( R R) and E(R R ) are protective (see [8]).…”