# P. B. Bhattacharya's Basic Abstract Algebra PDF

By P. B. Bhattacharya

This ebook gets you there if you happen to think in it. It has examples with strategies and issues of ideas. the one subject that doesn't have issues of suggestions is different types. For this, i've got the Hungerford textual content, and i'm shortly within the means of discovering a greater booklet for this. differently it's the excellent publication for self-study.

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**Example text**

An integer b is a factor or divisor of an integer a (or a is a multiple of b) if there exists an integer c such that a = bc. We say that b divides a and write bla (b%a means b does not divide a). A nonzero integer p is called a prime ifp ± I and the only factors of p are ± p and ± I. Let a be an integer. The absolute value (or modulus) IaI of a is defined as follows: a 0 1a ifa>O, ifa=0, ifa<0. The set of integers is denoted by Z, and the set of positive integers (also called natural numbers) by N.

A (strictly) lower triangular matrix is defined similarly. The set The set of all m X n matrices over a field F is denoted by of all square matrices of order n over F is, for the sake of conveof all nience, written F,,. Again, for the sake of convenience, the set I X n matrices (row vectors) and the set F"<' of all n X 1 matrices (col- umn vectors) over F are both written F" when the context makes the meaning unambiguous. Operations on matrices 41 Operations on matrices 2 In order to set up an algebraic system of matrices, we are going to intro- duce two binary operations, addition and multiplication, and a third operation called scalar multiplication.

We now show that (a,b) = (c,d) if and only if a = c and b = d. If a = c and b = d, then trivially (a,b) = (c,d). Conversely, let (a,b) = (c,d). Then ((a),(a,b)) = ((c),{c,d)). By definition of equality of sets, this implies (a) = (c) or (a) = (c,d). If (a) = (c), then we must have (a,b) = (c,d). This yields a = c, b = d. If, on the other hand, (a) = (c,d), then we must have (a,b) = (c). So a = c = d and a = b = c, which implies a = c = h = d. Definition. Let A,B be sets. The set of all ordered pairs (x,y).