# A b c r such that a 6 0 r and a is not a zero divisor

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a, b, c R such that a 6 = 0 R and a is not a zero divisor, then ab = ac implies b = c . Proof. ab = bc implies a ( b - c ) = 0 R . The fact that a 6 = 0 and a is not a zero divisor implies that we must have b - c = 0 R , i.e., b = c . 2 Theorem 5.4 If D is an integral domain, then 1. for all a, b, c D , a 6 = 0 D and ab = ac implies b = c ; 2. for all a, b D , a | b and b | a if and only if a = bc for c D * . Proof. The first statement follows immediately from the previous theorem and the definition of an integral domain. For the second statement, if a = bc for c D * , then we also have b = ac - 1 ; thus, b | a and a | b . Conversely, a | b implies b = ax for x D , and b | a implies a = by for y D , and hence b = bxy . Cancelling b , we have 1 D = xy , and so x and y are units. 2 It follows from the above theorem that in an integral domain D , if a, b D with b 6 = 0 D and b | a , then there is a unique c D such that a = bc , which we may denote as a/b . 5.1.3 Subrings A subset R 0 of a ring R is called a subring if R 0 is an additive subgroup of R , R 0 is closed under multiplication, i.e., ab R 0 for all a, b R , and 1 R R 0 . Note that the requirement that 1 R R 0 is not redundant. Some authors do not make this restriction. It is clear that the operations of addition and multiplication on R make R 0 itself into a ring, where 0 R is the additive identity of R 0 and 1 R is the multiplicative identity of R 0 . Example 5.10 Z is a subring of Q . 2 37

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5.1.4 Direct products of rings If R 1 , . . . , R k are rings, then the set of all k -tuples ( a 1 , . . . , a k ) with a i R i for 1 i k , with addition and multiplication defined component-wise, forms a ring. The ring is denoted R 1 ×· · ·× R k , and is called the direct product of R 1 , . . . , R k . Clearly, ( a 1 , . . . , a k ) is a unit (resp., zero divisor) in R 1 ×· · ·× R k if and only if each component a i is a unit (resp., zero divisor) in R i . 5.2 Polynomial rings If R is a ring, then we can form the ring of polynomials R [ T ], consisting of all polynomials k i =0 a i T i in the indeterminate (or variable) T , with coefficients in R , with addition and multipli- cation being defined in the usual way: let a = k i =0 a i T i and b = i =0 b i T i ; then a + b := max( k,‘ ) X i =0 ( a i + b i ) T i , where one interprets a i as 0 R if i > k and b i as 0 R if j > ‘ , and a · b := k + X i =0 c i T i , where c i := i j =0 a j b i - j , and one interprets a j as 0 R if j > k and b i - j as 0 R if i - j > ‘ . For a = k i =0 a i T i R [ T ], if k = 0, we call a a constant polynomial, and if k > 0 and a k 6 = 0 R , we call a a non-constant polynomial. Clearly, R is a subring of R [ T ], and consists precisely of the constant polynomials of R [ T ]. In particular, 0 R is the additive identity of R [ T ], and 1 R is the multiplicative identity of R [ T ]. 5.2.1 Polynomials versus polynomial functions Of course, a polynomial a = k i =0 a i T i defines a polynomial function on R that sends x R to k i =0 a i x i , and we denote the value of this function as a ( x ). However, it is important to to regard polynomials over R as formal expressions, and not to identify them with their corresponding functions. In particular, a polynomial a =
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