# Cyclic Modules and the Structure of Rings by S.K. Jain

By S.K. Jain

This targeted and accomplished quantity presents an updated account of the literature as regards to deciding on the constitution of jewelry over which cyclic modules or right cyclic modules have a finiteness situation or a homological estate. The finiteness stipulations and homological homes are heavily interrelated within the experience that both speculation induces the opposite in a few shape. this can be the 1st ebook to carry all of this crucial fabric at the topic together.

Over the final 25 years or extra a variety of mathematicians have investigated earrings whose issue jewelry or issue modules have a finiteness situation or a homological estate. They made very important contributions resulting in new instructions and questions, that are indexed on the finish of every bankruptcy for the advantage of destiny researchers. there's a wealth of fabric at the subject that is mixed during this publication, it includes greater than 2 hundred references and isn't claimed to be exhaustive.

This ebook will attract graduate scholars, researchers, and pros in algebra with a data of uncomplicated noncommutative ring thought, in addition to module thought and homological algebra, reminiscent of a one-year graduate path within the idea of earrings and modules.

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**Extra info for Cyclic Modules and the Structure of Rings**

**Sample text**

Since R/D ∼ = C/B, the module R/D is injective. Since B is countably generated and D/B is cyclic, D is countably generated. 2, D is ﬁnitely generated. Therefore D is a cyclic direct summand of RR . Since B is essential in RR , we have D = R. Therefore C/B = 0, a contradiction to our assumption. This shows that the module R/(A ⊕ (1 − e)R) does not contain a nonzero injective submodule. 4 (Osofsky’s Theorem) For a ring R, the following conditions are equivalent. 1. Each cyclic right R-module is injective.

So R contains independent uniform right ideals U1 , . . , Uk such that U = U1 ⊕ · · · ⊕ Uk ⊂e RR . Thus R/U is an artinian right R-module. Moreover, since R has right RM C, it is easy to check that for each nonzero submodule Vi of Ui , Ui /Vi is artinian. Hence U has Krull dimension at most 1. Thus R has right Krull dimension. 8, [180]). Therefore R has ACC on right annihilators. Hence R is quasi-Frobenius (see [70]). Case 2: J(R) is nil. 6, R/S has ﬁnite right Goldie dimension. Since R/S also has right RM C, we can use a similar argument to that given above to show that the ring R/S has right Krull dimension at most 1.

Rings of type (4) are restricted selfinjective since their proper homomorphic images have at most one proper ideal, hence are maximal rank 0 uniserial rings. Finally, let R be a domain of type (1). Then every nonzero prime ideal of R is maximal (since this is true of each RM ). Consequently, for each nonzero ideal A of R, the ring R/A has only a ﬁnite number of maximal ideals and all of its primes are maximal. It may be shown that R/A is a direct sum of a ﬁnite number of rings of the form RM /AM , where M a maximal ideal of R.