Read e-book online Abelian Groups and Modules PDF

By K.M. Rangaswamy, David Arnold

ISBN-10: 0824797892

ISBN-13: 9780824797898

Includes the court cases of a world convention on abelian teams and modules held lately in Colorado Springs. offers the most recent advancements in abelian teams that experience facilitated cross-fertilization of latest innovations from different components akin to the illustration idea of posets, version conception, set idea, and module concept.

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Extra resources for Abelian Groups and Modules

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If X 9^ 0 is a generating set of M then this means that every element m G M may be written as a finite linear combination n m = £ je//, JC/ € AT, r G i?. y ; = 1 It is here obvious that n G N is not fixed in general but depends on m. Further the coefficients r and, in fact, also the x - G X that occur are not uniquely determined by m. Of course if X = { * i , . . , x ] is finite then every element meM can be written in the form } } t t m = Z y-i since the missing summands xfi can be added as x,0.

The proof may be left to the reader. 5 FACTOR MODULES AND F A C T O R RINGS The definition of factor modules holds as in the case of factor spaces of linear vector Spaces since only properties of linearity are employed in the definition. Let C ^ M . Then, in particular, C is a subgroup of the additive group of M Clearly the factor group M/C = {m + C\meM} exists under the addition R (ma + C H O n z + C) := ( m i + m ) + C. 2 A module multiplication can now be defined on M/C so that M/C becomes a right module termed a factor module or a residue class module of M modulo C or also of M by C.

Obviously = is an equivalence relation in the class of all rings. A n isomorphism of R with R is called an automorphism. As for modules there exist ring homomorphisms i and v as well as 0, in the case that the zero ring is admitted. 2). Further it is clear that the image of a (unitary) subring with respect to a (unitary) ring homomorphism p is again a (unitary) subring of Cod(p). I n particular lm(p) is a (unitary) subring of Cod(p). For the most part the ideals of a ring are more important than the subrings.

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Abelian Groups and Modules by K.M. Rangaswamy, David Arnold

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