# Foundations of module and ring theory: A handbook for study by Robert Wisbauer

By Robert Wisbauer

This quantity offers a finished creation to module idea and the similar a part of ring idea, together with unique effects in addition to the newest paintings. it's a worthwhile and stimulating examine for these new to the topic in addition to for researchers and serves as an integral reference quantity.

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If R is division ring (arbitrary ring), then the {fa }a∈G form a basis of the vector space (free R-module) RG. 6 Remarks: (1) The embedding G → RG just constructed allows us to consider the elements of G as elements of RG and to write f ∈ RG as f = a∈G ra a, ra ∈ R zero almost everywhere. Therefore RG is also called the ring of formal linear combinations of elements of G with coefficients in R. 2). These are 1 or 0 according to the multiplication table of G. (3) Recall that, for a commutative ring R and G = (IN , +), using the above notation an indeterminate X = f1 is defined by f1 : R → RIN = R[X], f1 (r) = δ1,r , r ∈ R.

For f ∈ Mor(A, B), we call A the source of f , B the target of f , and we f write f : A → B or A −→ B. 44 Chapter 2 Module categories For A, B, C, D ∈ Obj(C), the morphisms f ∈ MorC (A, B), g ∈ MorC (B, D), h ∈ MorC (A, C), k ∈ MorC (C, D) can be presented in the following diagram: f A h↓ −→ C −→ k B ↓g D The diagram is commutative if f g = hk. Remark: The definition of a category is based on a class of objects. e. ∈). Every set is a class. The totality of all sets forms a class but not a set. Two classes are equal if they contain the same elements.

Def initions. Let C be a category with zero object and f : A → B a morphism in C. e. we have a commutative diagram D ↓g h K i → A f → B. e. we have the commutative diagram f A → B g↓ D p → C h . e. e. p : B → C) is an epimorphism. (2) The source of a kernel of f and the target of a cokernel of f are uniquely determined up to isomorphisms, more precisely: If i1 : K1 → A, i2 : K2 → A are two kernels of f, then there is an isomorphism α : K1 → K2 with i1 = αi2 . A similar result holds for the cokernel of f.