Divisibility in Integral Domains
Divisibility in Integral Domains Recall: A commutative ring with unity D is called an integral domain if D contains no zero divisors. That is, if a, b D and ab =0,wehaveeithera =0or b =0. 2 Recall: An element x of a ring R is called a unit if there is an element y in R such that xy = yx =1,where1isunityinR. Some examples of integral domains: 1. All fields, such as Q, R, C,andZp where p is prime 2. The ring of integers Z 3. The ring of polynomials D[x] if D is an integral domain 4. The Gaussian integers, a + bi : a, b Z { 2 } Definition Let D be an integral domain. If a, b D,wesaythata is a divisor of b,orthatb is a multiple of a,orthata b,ifthere is an element x in D such that ax =2b. | Example In Z, 3 is a divisor of 18, but 4 is not, since there is no integer x such that 4x = 18. NOTE THE DIFFERENCE FROM OUR USUAL DEFINITION: Example In R, 3 and 4 are both divisors of 18, since there are real numbers x and y such that 3x = 18 and 4y = 18. Definition Let D be an integral domain. If a, b D and a = ub,whereu is a unit in D,thena and b are called associates. Equivalently, a and b are associates if both2 a b and b a. | | Example In Z,1and-1areassociates. Definition A nonzero element a in a ring R is called an irreducible if a is not a unit and if whenever a = bc,theneitherb or c is a unit in R.Equivalently,ifa is an irreducible if whenever a = bc,theneithera b or a c.
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