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Unformatted text preview: MATH 8020 CHAPTER 1: COMMUTATIVE RINGS PETE L. CLARK Contents 1. Commutative rings 1 1.1. Fixing terminology 1 1.2. Adjoining elements 4 1.3. Ideals and quotient rings 5 1.4. The monoid of ideals of R 8 1.5. Pushing and pulling ideals 9 1.6. Maximal and prime ideals 10 1.7. Products of rings 10 1.8. Additional exercises 13 1. Commutative rings 1.1. Fixing terminology. We are interested in studying properties of commutative rings with unity . 1 So let us begin by clarifying this terminology. By a general algebra R , we mean a triple ( R, + , · ) where R is a set endowed with a binary operation + : R × R → R – called addition – and a binary operation · : R × R → R – called multiplication – satisfying the following: (CG) ( R, +) is a commutative group, (D) For all a, b, c ∈ R , ( a + b ) · c = a · c + b · c, a · ( b + c ) = a · b + a · c . For at least fifty years, there has been agreement that in order for an algebra to be a ring , it must satisfy the additional axiom of associativity of multiplication: (AM) For all a, b, c ∈ R , a · ( b · c ) = ( a · b ) · c . A general algebra which satisfies (AM) will be called simply an algebra . A similar convention that is prevalent in the literature is the use of the term nonassociative algebra to mean what we have called a general algebra: i.e., a not necessarily 1 In particular, all rings considered in these notes are meant to be commutative. Any references to possibly non-commutative rings are vestiges of an older version of these notes. If you find them, please point them out to be so they can be removed. 1 2 PETE L. CLARK associative algebra. A ring R is said to be with unity if there exists a multiplicative identity, i.e., an element e of R such that for all a ∈ R we have e · a = a · e = a . If e and e ′ are two such elements, then e = e · e ′ = e ′ . In other words, if a unity exists, it is unique, and we will denote it by 1. A ring R is commutative if for all x, y ∈ R, x · y = y · x . In these notes we will be working always in the category of commutative rings with unity. In a sense which will be made precise shortly, this means that the identity 1 is regarded as a part of the structure of a ring, and must therefore be preserved by all homomorphisms of rings. Probably it would be more natural to study the class of possibly non-commutative rings with unity, since, as we will see, many of the fundamental constructions of rings give rise, in general, to non-commutative rings. But if the restriction to commutative rings (with unity!) is an artifice, it is a very useful one, since two of the most fundamental notions in the theory, that of ideal and module, become significantly different and more complicated in the non-commutative case. It is nevertheless true that many individual results have simple analogues in the non- commutative case. But it does not seem necessary to carry along the extra general- ity of non-commutative rings; rather, when one is interested in the non-commutative...
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