On I-finite left quasi-duo rings

Authors : Ayman M A HOROUB, W K NICHOLSON

Pages : 161-202

Doi:10.24330/ieja.1058427

View : 19 | Download : 11

Publication Date : 2022-01-17

Article Type : Research Paper

Abstract :A ring is called left quasi-duo insert ignore into journalissuearticles values(left QD); if every maximal left ideal is a right ideal, and it is called I-finite if it contains no infinite orthogonal set of idempotents. It is shown that a ring is I-finite and left QD if and only if it is a generalized upper-triangular matrix ring with all diagonal rings being division rings except the lower one, which is either a division ring or it is I-finite, left QD and left `soclin` insert ignore into journalissuearticles values(left QDS);. Here a ring is called left soclin if each simple left ideal is nilpotent. The left QDS rings are shown to be finite direct products of indecomposable left QDS rings, in each of which 1 = f 1 + ⋯ + f m 1=f1+⋯+fm where the f i fi are orthogonal primitive idempotents, with f k ≈ f l fk≈fl for all k , l , k,l, and ≈ ≈ is the block equivalence on { f 1 , … , f m } . {f1,…,fm}. A ring is shown to be left soclin if and only if every maximal left ideal is left essential, if and only if the left socle is contained in the left singular ideal. These left soclin rings are proved to be a Morita invariant class; and if a ring is semilocal and non-semisimple, then it is left soclin if and only if the Jacobson radical is essential as a left ideal . . Left quasi-duo elements are defined for any ring and shown to constitute a subring containing the centre and the Jacobson radical of the ring. The `width` of any left QD ring is defined and applied to characterize the semilocal left QD rings, and to clarify the semiperfect case . .
Keywords : Left quasi duo ring, left soclin ring, generalized triangular matrix ring, left max ideal, ideal simple module, left socle, very semisimple module