By Donald S. Passman

First released in 1991, this ebook includes the middle fabric for an undergraduate first path in ring thought. utilizing the underlying subject of projective and injective modules, the writer touches upon numerous features of commutative and noncommutative ring thought. particularly, a few significant effects are highlighted and proved. half I, 'Projective Modules', starts with easy module idea after which proceeds to surveying a number of precise sessions of jewelry (Wedderbum, Artinian and Noetherian earrings, hereditary earrings, Dedekind domain names, etc.). This half concludes with an creation and dialogue of the ideas of the projective dimension.Part II, 'Polynomial Rings', reviews those jewelry in a mildly noncommutative surroundings. a few of the effects proved contain the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, 'Injective Modules', contains, specifically, numerous notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The publication comprises a number of routines and an inventory of prompt extra studying. it really is appropriate for graduate scholars and researchers attracted to ring idea.

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First released in 1991, this e-book comprises the middle fabric for an undergraduate first path in ring concept. utilizing the underlying subject matter of projective and injective modules, the writer touches upon quite a few facets of commutative and noncommutative ring conception. particularly, a couple of significant effects are highlighted and proved.

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S be rings and assume that Rs is a finitely generated Smodule. If S is Artinian, prove that R is also. In particular, if R = Mn ( S) deduce that S is Artinian if and only if R is Artinian. 7. Suppose K ~ F are fields with dimK F' = oo and let R be the subring of M2 (F) given by R = ( ~ ~)· Show first that I= (~ ~)is a minimal right ideal of R and then construct a composition series for RR· Deduce that R is right Artinian and then prove that R is not left Artinian. 8. Show that there is a one-to-one correspondence between idempotents e E EndR(V) and direct sum decompositions V = X Y.

N - 1 and j = O, 1, ... , m - 1. , 26 Part I. Projective Modules since Xi+1 n YJ+1 ~ XH1 n Yj and Xi n (XH1 n YJ+1) =Xi n YJ+1· But note that the final expression for Xi,H1/Xi,j is symmetric in Xi and lj. Thus if we define Yi,j similarly by then Xi,j+1/ Xi,j equivalent. ~ Yi+i,j /Yi,j and therefore the two refinements are D A composition series for Vis a series O=Vo~Vi~···~Vn=V such that each factor Vi+1/Vi is irreducible. Needless to say, not every module has such a series. 5 (Jordan-Holder Theorem) Any two composition series for an Rmodule V are equivalent, that is they have the same length and isomorphic factors.

If (1 - x)R ':/: R then, by Zorn's Lemma, we have (1- x)R ~ M for some maximal right ideal M. But I ~ Rad(R) ~ M, so this implies that both x and 1 - x are in M, clearly a contradiction. We conclude that (1- x)R = R for all x E I and, in particular, that each such 1 - x. has a right inverse. Again let x E I and let 1 - y be a right inverse for 1 - x. Then 1 = (1 - x) (1 - y), so y = xy - x E I and thus 1 - y also has a right inverse. But then 1 - y has both a left and a right inverse, so it follows that 1-y is invertible with inverse 1- x.