By Yurij A. Drozd, Vladimir V. Kirichenko (auth.)
This English variation has an extra bankruptcy "Elements of Homological Al gebra". Homological tools seem to be powerful in lots of difficulties within the conception of algebras; we are hoping their inclusion makes this ebook extra whole and self-contained as a textbook. we now have additionally taken this party to right a number of inaccuracies and error within the unique Russian variation. we must always prefer to convey our gratitude to V. Dlab who has not just metic ulously translated the textual content, yet has additionally contributed by means of writing an Appendix dedicated to a brand new very important category of algebras, viz. quasi-hereditary algebras. eventually, we're indebted to the publishers, Springer-Verlag, for allowing this booklet to arrive this kind of extensive viewers on this planet of mathematical group. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface the speculation of finite dimensional algebras is among the oldest branches of contemporary algebra. Its beginning is associated with the paintings of Hamilton who chanced on the well-known algebra of quaternions, and Cayley who constructed matrix concept. Later finite dimensional algebras have been studied via a lot of mathematicians together with B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. on the finish of the final century T. Molien and E. Cartan defined the semisimple algebras over the advanced and actual fields and paved the 1st steps in the direction of the examine of non-semi uncomplicated algebras.
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Additional info for Finite Dimensional Algebras
Let M be a cyclic module over an algebra A, generated by m. Prove that M ~ AI Ann m. Verify that, if Ann m is an ideal of A (for instance in case that A is commutative), then Ann m = Ann AI and for an arbitrary generator m' of M, Annm' = Annm. Show that, if A is not commutative, then Ann m may depend on the choice of m. ) 10. (Peirce decomposition of an ideal) Let I be an ideal of an algebra A and 1 = el + e2 + ... + e s a decomposition of the identity of this algebra. Prove that the element a = L aij, where aij E eiAej, belongs to I if and only if aij E e;l ej.
But A is a semisimple algebra, and thus J I R = 0, i. e. J = R. 8 provides a symmetric characterization of the radical (free of the concepts "right" and "left"). It follows that the radical of the left regular module coincides with the radical of the algebra. 9. The radical R of an algebra A is a nilpotent ideal containing all nilpotent right and left ideals. Proof. 5) implies that R2 = radR # R and, in general, Rm+l = radRm # R m whenever Rm # O. However, the chain of the subspaces R :::> R2 :::> ••• :::> Rm:::> Rm+l :::> ••• must become stationary, and thus R m = 0 for some m.
End(V). a) Define a module over an algebra without an identity and establish a relation between the modules and representations. b) Let M be a module over an algebra A without an identity. Put m(a,a) = ma + am (a E A, a E J(). Prove that M is an A-module. i(M,N), considering M and N as A-modules (see b». In particular, M ~ N (as A-modules) if and only if M ~ N considered as A-modules. 2. Semisimple Algebras The classical theory of semisimple algebras is one of the most striking examples how "module theoretical" methods produce deep structural results.
Finite Dimensional Algebras by Yurij A. Drozd, Vladimir V. Kirichenko (auth.)