By Thomas C. Craven

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**Sample text**

In the former case, we write inverses as −a and e as 0. In the latter we write inverses as a−1 , but e is still the usual name for the identity. Examples: we have already seen essentially all the abelian finite groups. 7 says they are just the additive subgroups of our rings Zn . In fact, a little more can be said because these rings are often products (recall Z6 ∼ = Z2 × Z3 ), but there is not too much more to learn about these. That means we have to start dealing much more with noncommutativity!

B) It was surjective in the example mentioned above: Z6 → Z6 /(3) × Z6 /(2) ∼ = Z2 × Z3 . On the other hand, for Z → Z/(2) × Z/(4), nothing maps onto (1, 0) since n ≡ 1 (mod 2) implies n ≡ 0 (mod 4). (c) Check that ker f = I ∩ J. The Chinese Remainder Theorem in Chapter 13 deals with the issue of when the mapping is surjective. Prime ideals. What is special about an ideal I for which R/I is an integral domain or a field? For this section we assume that R is a commutative ring with identity, since these are necessary conditions to hope to have an integral domain.

N }, a permutation of S is just a bijective function f : S → S. Since compositions of bijections are again bijective, the identity is a bijection, and bijections have inverses, the set of all permutations of S form a group. We will call the group Sn , the symmetric group on n symbols. , as you saw in linear algebra. We shall study these groups in considerable detail next semester. For n = 3, the details are all written out in the book, pp 161–163. We shall later use a more compact notation which will help us understand the patterns involved in permutations (cycles).