Download Algebra for Computer Science by Lars Garding, Torbjörn Tambour PDF

By Lars Garding, Torbjörn Tambour

The target of this e-book is to coach the reader the themes in algebra that are important within the research of computing device technological know-how. In a transparent, concise kind, the writer current the fundamental algebraic buildings, and their functions to such themes because the finite Fourier remodel, coding, complexity, and automata thought. The ebook is usually learn profitably as a direction in utilized algebra for arithmetic students.

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We begin with a LEMMA. A submodule of a finitely generated free module is free. PROOF: Let A be a finitely generated free module and B a submodule. We will use induction over the number of generators of A. We leave it to the reader to verify that a submodule of a free module generated by one element is free. Let aI, ... ,ak be a set of generators of A such that Define a morphism p : A - Zak by The kernel B' of the restriction of p to B is contained in hence is free by the induction hypothesis. If p(B) = 0, then we are finished.

From the abstract point of view there is no difference between a module and a commutative group. We just have to write addition and subtraction instead of multiplication and division and vice versa. 1 The four operations of arithmetic 37 multiplicative unit and the zero for addition correspond to each other. The distinction between the two concepts is just traditional and terminologically convenient. Number theory and arithmetic mod m offer a non-trivial example of a commutative ring, namely the set Zm of congruence classes C(z) = z+mZ mod m.

Zak. = = Then F is free. In fact, if n1a1 + ... + nkak 0, then n1a1 + ... + nkak 0 and n1 = ... = nk = O. We claim that A is the direct sum of F and T(A). For if a is in A, then a = n1a1 + ... + nkak for some integers nl, ... ,nk. Hence a - n1a1 - ... - nkak E T(A) since its image in A' vanishes. It remains to prove that F n T(A) = O. Suppose that a E F n T(A). Write a = n1a1 +. +nkak. Then 0 = a = n1a1 +. ·+nkak and n1 = ... = nk = 0 since A' is free. Hence a = O. The proof is finished. R. Prove that T(A) is finite if A is finitely generated.

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