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

By S.K. Jain

This particular and complete quantity offers an up to date account of the literature with regards to selecting the constitution of earrings over which cyclic modules or right cyclic modules have a finiteness or a homological estate. The finiteness stipulations and homological houses are heavily interrelated within the experience that both speculation induces the opposite in a few shape. this is often the 1st ebook to carry all of this crucial fabric at the topic together.

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

This ebook will attract graduate scholars, researchers, and pros in algebra with an information of uncomplicated noncommutative ring thought, in addition to module conception and homological algebra, corresponding to a one-year graduate path within the thought of jewelry and modules.

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**Example text**

This gives us a ring as described in (3). = 0 eRe The converse is obvious when R is a ring of type (1) or (2). So suppose now AM that R is a ring of type (3), that is, R = , where A, B are simple artinian 0 B rings and M is an irreducible (A; B)-bimodule. Since R/J(R) is von Neumann regular, it is enough to show that each nonzero ideal I of R contains J(R). Let p ∈ I. Then p = ae11 + me12 + be22 , where a ∈ A, b ∈ B and m ∈ M . We have a e11 pb e22 = a mb e12 ∈ I, for all a ∈ A, b ∈ B. Thus AmBe12 ⊂ I.

From this, we conclude that S is a semiprimary ring. It follows that S is an artinian ring since S has left and right restricted minimum condition. Therefore R is a noetherian ring. A ring R is said to satisfy the restricted right socle condition (RRS for short) if for each essential right ideal E (= R), R/E has nonzero socle. A ring R is called a right QI-ring if each quasi-injective right R-module is injective. Boyle proved that a right QI-ring is right noetherian, and conjectured that every right QI-ring is hereditary [30].

Thus R is semiprimary. It follows that R/S is a semiprimary ring with right Krull dimension, so R/S is right artinian. Therefore, R has DCC on essential right ideals, and hence R is quasi-Frobenius (see [14]). Chatters [44] studied a hereditary noetherian ring R which satisﬁes the restricted minimum condition and proved the following. 10 Let R be a left noetherian left hereditary ring, and let I be a ﬁnitely generated essential right ideal of R. Then R satisﬁes the descending chain condition for ﬁnitely generated right ideals which contain I.