Finitely Generated Abelian Groups

All subgroups K of a free Abelian group G are normal as it is Abelian. It is clear from Theorem 3.16 that G K is finitely generated: K eliminates gener- ators x i of G when d i = 1 and turns others into generators with finite order d i 1. This statement extends to finitely generated groups, as their subgroups are finitely generated and a similar factorization occurs. The corollary follows. Corollary 3.3 Let G be a finitely generated Abelian group with free part of rank n. Let K be a subgroup of G with free part of rank s ≤ n. Then, GK is finitely generated and its free part has rank n − s. Example 3.7 factoring finitely generated groups Theorem 3.10 factors a finitely generated Abelian group as the product of a free Abelian group and a number of finite cyclic groups. Using Theorem 3.14, we may restate the result of Theorem 3.10: Every finitely generated Abelian group G may be fac- tored into a free Abelian group H and the product of finite cyclic groups T , G = H × T . Then, GT ∼ = H ∼ = Z β , where β is the Betti number of G. T ∼ = T is often called the torsion subgroup of G, and it contains all generators with finite orders.

3.3.2 Rings, Fields, Integral Domains, and Principal Ideal Domains

The concepts of bases and ranks are familiar to most readers from basic linear algebra and vector spaces. There is, indeed, a direct connection, which we will unveil next. We begin by allowing two binary operations for a set. Definition 3.20 ring with unity A ring R, +, · is a set R together with two binary operations + and ·, which we call addition and multiplication, de- fined on R such that the following axioms are satisfied: a R, + is an Abelian group. b Multiplication is associative. c For a , b, c ∈ R, the left distributive law, ab + c = ab + ac, and the right distributive law , a + bc = ac + bc, hold. A ring R with a multiplicative identity 1 such that 1x = x1 = x for all x ∈ R is a ring with unity. Definitions and concepts from groups naturally extend to rings, sometimes with different names. Rather than defining them individually, I list the equiv- alent concepts in Table 3.3. For example, a ring with a commutative multipli- cation operation is called a commutative ring. Using this table, we now define fields, the richest most restrictive structure we will encounter. Table 3.3. Equivalent concepts for groups and rings. groups rings Abelian commutative subgroup subring normal ideal cyclic principal Definition 3.21 field A field F is a commutative ring with unity such that, for all a ∈ F, there is an element a − 1 such that aa − 1 = a − 1 a = 1. In other words, multiplicative inverses exist in fields. A sibling structure of a field is an integral domain, where the elements do not necessarily have multi- plicative inverses. Definition 3.22 integral domain An integral domain D is a commutative ring with unity such that, for all nonzero a , b ∈ D, ab = 0. An integral domain captures the properties of the set of integers in abstract algebra, hence the name. Other concepts from the set of integers carry over as well. Definition 3.23 unit, irreducible An element u of an integral domain D is a unit of D if u has a multiplicative inverse in D. A nonzero element p ∈ D that is not a unit of D is an irreducible of D if in any factorization p = ab in D either a or b is a unit. So, the concept of primes in Z is generalized to the concept of irreducibles for any integral domain. Fields and integral domains are very much related. Theorem 3.17 Every field is an integral domain. Every finite integral domain is a field. Example 3.8 Z , Q, R, C are all rings under the operations of addition and mul- tiplication. Z n , +, · n is a ring where · n is multiplication modulo n. Z is not a field, because it does not have multiplicative inverses for its elements, but Z is an integral domain. Q and R are fields, and therefore integral domains. Z p is an integral domain if p is prime. As Z p is finite, Theorem 3.17 implies that Z is also a field. If p is not a prime, Z p is not an integral domain, as it has nonzero elements that divide zero. For example, 2 · 6 3 = 0 in Z 6 .