By Lars Garding, Torbjörn Tambour
The objective of this e-book is to coach the reader the themes in algebra that are invaluable within the examine of desktop technology. In a transparent, concise variety, the writer current the elemental algebraic constructions, and their functions to such issues because the finite Fourier remodel, coding, complexity, and automata thought. The booklet is additionally learn profitably as a path in utilized algebra for arithmetic students.
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Extra info for Algebra for Computer Science
Otherwise = = = O,a, ... ,(m-l)a are all the elements of Za and they are all different. Addition and subtraction in Za are performed under the condition that ma = O. This determines these operations completely; we have ra = sa if and only if r == s (m). This module is said to be cyclic of order m , where 'order' means the number of elements. When a = 0, Za contains only the element O. The elements of a cyclic module of order m behaves exactly as the congruence classes mod m under addition and subtraction.
Prove also that if a and b are the generators of A and B, then a + b has order mn. Prove that (ra, sb) -+ (rm' + sn')(a + b) where mm' + nn' = 1 is an isomorphism from A $ B to the cyclic module generated by a + b. Exercise Let M and N be modules. Show that the direct sums M $ Nand N $ M are isomorphic. Let 1:M -+ M' be a module morphism. Then the image iml = I(M) of 1 is a submodule of M'. To see this, note that if I(a) and I(b) are images of elements in M, so is I(a) - I(b) = I(a - b). The set of eleme~ts in M which are mapped by 1 to the zero 0' of M' is called the kernel of I, ker I.
PROOF: The elements of A + B have the form a + b with a in A and b in B. Hence A + B is a module. Let a be the generator of A and b that of B. Then c = a + b has the order mn. In fact, if rc = 0, then 0 = ra + rb so that 0 = mra = -mrb. It follows that n divides r. Similarly, m divides r. Hence r is a multiple of mn which is also the order of c. Since A + B has at most mn elements, c generates A + B. This finishes the proof. Group notation For commutative groups, the notion of submodule corresponds to subgroup.