Ideal intrinsic extensions with connections to PI-rings

Abstract. Recall that an n-by-n generalized matrix ring is defined in terms of sets of rings {R i } n i=1 , (R i , R j )-bimodules {M ij }, and bimodule homomorphisms θ ijk : M ij ⊗ Rj M jk → M ik , where the set of diagonal matrix units {E ii } form a complete set of orthogonal idempotents. Moreover, an arbitrary ring with a complete set of orthogonal idempotents {e i } n i=1 has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring R π which is isomorphic to R. In this paper, we focus on the subclass T n of n-by-n generalized matrix rings with θ iji = 0 for i = j. T n contains all upper and all lower generalized triangular matrix rings. The triviality of the bimodule homomorphisms motivates the introduction of three new types of idempotents called the inner Peirce, outer Peirce, and Peirce trivial idempotents. These idempotents are our main tools and are used to characterize T n and define a new class of rings called the n-Peirce rings. If R is an n-Pierce ring, then there is a certain complete set of orthogonal idempotents {e i } n i=1 such that R π ∈ T n . We show that every n-by-n generalized matrix ring R contains a subring S which is maximal with respect to being in T n and S is essential in R as an (S, S)-bisubmodule of R. This allows for a useful transfer of information between R and S. Also, we show that any ring is either an n-Peirce ring or for each k > 1 there is a complete set of orthogonal idempotents {e i } k i=1 such that R π ∈ T k . Examples are provided to illustrate and delimit our results.

show abstract

“…( 3) and (4). By Proposition 2.4 we have [ABP,Theorem 1.6(i)], and (4) follows from [ABP,Theorem 1.6(ii)].…”

Section: Characterization Of T Nmentioning

confidence: 90%