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Unformatted text preview: 25. Ideals and quotient rings We continue our study of rings by making analogies with groups. The next concept we introduce is that of an ideal of a ring. Ideals are ringtheoretic counterparts of normal subgroups. Recall that one of the main reasons why normal subgroups are important is that they can be used to construct quotient groups. Similarly, ideals are special kinds of subrings, and at the end of the lecture we will see that to each ideal of a ring, one can associate a quotient ring. While ideals can be defined in arbitrary rings, to simplify the matters we will only consider ideals in commutative rings; in fact, in all examples we will deal with commutative rings with 1. 25.1. Ideals. Definition. Let R be a commutative ring and I a subset of R . Then I is called an ideal if (a) I is a subgroup of ( R, +) (i.e., a subgroup with respect to addition) (b) I absorbs products with R . This means that for any x I and r R we must have xr I . Remark: Note that this definition does not explicitly say that an ideal must be a subring. This, however, is an easy consequence of the definition, as we will see shortly. Example 1: Let R = Z , fix n Z , and let I = n Z (the set of all integer multiples of n ). Then I is an ideal of R . Indeed, we already know that n Z is a subgroup of ( Z , +), and n Z clearly satisfies the product absorption condition (b) (if x is a multiple of n and r is any integer, then xr is also a multiple of n ). In fact, Example 1 is a special case of the more general Example 2: Example 2: Let R be any commutative ring with 1, fix a R , and let I = aR = { ar : r R } , that is, I is the set of all multiples of a . Then I is an ideal of R , called the principal ideal generated by a . Sometimes aR is denoted by ( a ). Let us prove that I = aR is an ideal: (a) First we prove that I is a subgroup of ( R, +). This, in turn, boils down to the following three conditions: 1 2 (i) I contains 0 (ii) I is closed under addition (iii) I is closed under additive inversion (i) holds since 0 = a aR = I ....
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 Winter '11
 PhanThuongCang
 Normal subgroup, Ring, Abelian group, R R

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