4400algebra2 - ALGEBRA HANDOUT 2 IDEALS AND QUOTIENTS PETE...

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ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!) ring. An ideal of R is a subset I of R satisfying: (IR1) I is a subgroup of the additive group of R . (IR2) For any r R and any i I , ri I . We often employ notation like rI = { ri | i I } and then (IR2) can be stated more succinctly as: for all r R , rI I . In other words, an ideal is a subset of a ring R which is a subgroup under addition (in particular it contains 0 so is nonempty) and is not only closed under multiplication but satisfies the stronger property that it “absorbs” all elements of the ring under multiplication. Remark (Ideals versus subrings): It is worthwhile to compare these two notions; they are related, but with subtle and important differences. Both an ideal I and a subring S of a ring R are subsets of R which are subgroups under addition and are stable under multiplication. However, each has an additional property: for an ideal it is the absorption property (IR2). For instance, the integers Z are a subring of the rational numbers Q , but are clearly not an ideal, since 1 2 · 1 = 1 2 , which is not an integer. On the other hand a subring has a property that an ideal usually lacks, namely it must contain the unity 1 of R . For instance, the subset 2 Z = { 2 n | n Z } is an ideal of Z but is not a subring. Example 1 (trivial ideals): Any ring R (which is not the zero ring!) contains at least two ideals: the ideal { 0 } , and the ideal R itself. These are however not very interesting examples, and often need to be ignored in a discussion. (The conven- tion that “ideal” should stand for “non-zero ideal” whenever convenient is a fairly common and useful one in the subject.) An ideal I is said to be proper if it is not R , and again most interesting statements about ideals should really be applied to proper ideals. Note well that an ideal is proper iff it does not contain the unity 1. Indeed, an ideal lacking 1 is certainly proper, and conversely, if 1 I and r R , then r · 1 = r is in I . Proposition 1. The following are equivalent for a nonzero commutative ring R : a) R has only the trivial ideals { 0 } and R . b) R is a field. Thanks to Chris Pryby for pointing out a typo in these notes. 1
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2 PETE L. CLARK Proof: b) = a): Suppose I is a nonzero ideal of a field R , so I contains some 0 = a . Then since a is a field, a - 1 exists and 1 = a - 1 a R · I I , so I contains 1 and is hence all of R . a) = b): Suppose R is not a field; then some nonzero element a does not have an inverse. Then the set aR = { ar | r R } is a proper, nonzero ideal. The preceding argument shows a general construction of ideals: for any element a of R , the set of elements { ra | r R } is an ideal of R . (Just to check: r 1 a + r 2 a = ( r 1 + r 2 ) a , - ra = ( - r ) a and r ( ra ) = ( r r ) a .) We denote such ideals as either Ra or ( a ); they are simple and easy to understand and are called principal ideals and a is called a generator .
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