A Generalization of the Theory of Standardly Stratified Algebras I: Standardly Stratified Ringoids

Abstract

We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

Standardly stratified lower triangular $\mathbb{K}$-algebras with enough idempotents

“…In [13], the authors introduce the notion of standardly stratified ringoid which is a generalization of the classical notion of standardly stratified algebra for semi-primary rings with unity. Moreover, they study the module category of standardly stratified ringoids (in particular, algebras with enough idempotents) and show that, even though they have no unity and an infinite number (up to isomorphisms) of simple modules, many classic results about the ∆-filtered representations of standardly stratified algebras can be generalized to representations of standardly stratified ringoids.…”

“…It is also remarked that certain equivalent characterizations of standardly stratified algebras and quasi-hereditary algebras are not necessarily equivalent any more in the realm of ringoids. The theory developed in [13] is then specified to algebras (over commutative rings) which have enough idempotents but need not have an unity. As a result, the notions of standardly stratified algebra and quasihereditary algebra are expanded in [13] to algebras which may not have an unity.…”

“…The theory developed in [13] is then specified to algebras (over commutative rings) which have enough idempotents but need not have an unity. As a result, the notions of standardly stratified algebra and quasihereditary algebra are expanded in [13] to algebras which may not have an unity.…”

“…In this paper we study the lower triangular matrix K-algebra Λ := [ T 0 M U ] , where U and T are basic K-algebras with enough idempotents and M is an U -T -bimodule where K acts centrally. The motivation, for doing so, is to generalize to the setting of [13] the results obtained by E. Marcos, H. Merklen and C. Sáenz in [11], where they studied the lower triangular matrix algebra Λ for U and T finite dimensional left standardly stratified algebras with unity. It is worth mentioning that some of the results in [11] where also obtained, independently, by B. Zhu for the matrix algebra Λ where U and T are quasihereditary algebras with unity [19].…”

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In this paper we study the lower triangular matrix K-algebra Λ := T 0 M U , where U and T are basic K-algebras with enough idempotents and M is an U -T -bimodule where K acts centrally. Moreover, we characterise in terms of U, T and M when, on one hand, the lower triangular matrix K-algebra Λ is standardly stratified in the sense of [13]; and on another hand, when Λ is locally bounded in the sense of [3]. Finally, it is also studied several properties relating the projective dimensions in the categories of finitely generated modules mod(U ), mod(T ) and mod(Λ).

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