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).

中文翻译：

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标准层代数理论概论 I：标准层状环

我们扩展了标准分层的经典概念ķ-代数（表示有限维ķ-algebras) 到更一般的环类，可能没有 1，具有足够的幂等性。我们表明，许多以经典标准分层代数而闻名的基本结果可以推广到这种情况。此外，新的环类表现为：理想的标准分层和理想的准遗传。在经典理论中，已知准遗传代数和理想准遗传代数是等价的概念，但在我们的一般环境中，这不再是正确的。为了发展该理论，我们使用了具有足够幂等性的环和骨架小类别（具有多个对象的环或环）之间的众所周知的联系。