Prime Goldie Rings of Uniform Dimension at Least Two and with All One-Sided Ideals CS Are Semihereditary

Huynh, Dinh Van; Jain, S. K.; López-Permouth, Sergio R.

doi:10.1081/agb-120023960

Cited by 4 publications

(1 citation statement)

References 11 publications

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“…On the other hand, if there is a right ideal A for which there are i # j such that At © Aj is quasi-continuous, then A t and A, are mutually injective by [13 [11] that every nonzero left (right) ideal of R is CS and so is a direct sum of uniform left (right) ideals. This shows that R is a ring satisfying (a) of Theorem 3.1.…”

Section: Simple Rings and Quasi-continuous Right Idealsmentioning

confidence: 99%

Simple rings with injectivity conditions on one-sided ideals

Clark¹,

Huynh²

2007

Bull. Austral. Math. Soc.

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This paper looks at simple rings which have right ideals satisfying various types of injectivity conditions. We characterise when a simple regular ring is right self-injective and show that if R is a simple ring in which every right ideal is the direct sum of quasi-continuous right ideals then R is either Artinian or a non-selfinjective right Goldie ring in which every right ideal is a direct sum of uniform right ideals. Moreover, a module M is called continuous if it is CS and every submodule of M which is isomorphic to a direct summand of M is in fact a direct summand of M. As usual, we say that the ring R is right continuous if the module RR is continuous. Any injective module is continuous and every continuous module is quasi-continuous but the converses fail in general (see [13,6]).Next, given two modules M and N, M is said to be N-injective (or M is injective relative to N) if, for each monomorphism f : K -¥ N and homomorphism h : K -¥ M,

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Section: Simple Rings and Quasi-continuous Right Idealsmentioning

confidence: 99%