305 -Rings, ideals, homomorphisms (3)

March 21, 2009

In order to understand the construction of the quotient ring from last lecture, it is convenient to examine some examples in details. We are interested in ideals {I} of {{\mathbb F}[x],} where {{\mathbb F}} is a field. We write {{\mathbb F}[x]/I} for the quotient ring, i.e., the set of equivalence classes {[a]_\sim} of polynomials {a} in {F[x]} under the equivalence relation {a\sim b} iff {a-b\in I.}

  • If {I=\{0\},} then for any {a,} the equivalence class {[a]_\sim} is just the singleton {\{a\}} and the homomorphism map {h:{\mathbb F}[x]\rightarrow{\mathbb F}[x]/I} given by {h(a)=[a]_\sim} is an isomorphism.

To understand general ideals better the following notions are useful; I restrict to commutative rings with identity although they make sense in other contexts as well:

Definition 1 Let {R} be a commutative ring with identity. An ideal {I} is principal iff it is the ideal generated by an element {a} of {R,} i.e., it is the set {(a)} of all products {ab} for {b\in R.}

For example, {\{0\}=(0)} is principal. In {{\mathbb Z}} every subring is an ideal and is principal, since all subrings of {{\mathbb Z}} are of the form {n{\mathbb Z}=(n)} for some integer {n.}

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305 -Algebra and induction (2)

January 23, 2009

Last time we showed that given (integers) m,n with n>0, there are q,r with 0\le r<n such that m=nq+r. We began today by showing that these integers q,r are unique. When r=0, we say that n divides m, in symbols n\mid m.

Definition. A greatest common divisor of the integers m,n not both zero, is a positive integer d that divides both m,n and such that any integer that divides both m,n, also divides d.

We showed that for any m,n not both zero, there is a unique such d, in symbols d={\rm gcd}(m,n) or simply d=(m,n). We also showed the following characterization:

Theorem. Let m,n be integers, not both zero. Let S=\{l\in{\mathbb Z}^+: for some integers x,y, l=mx+ny\}.  Then the following are equivalent statements about the integer d:

  1. d=(m,n).
  2. d \mid m, d\mid n, and d\in S.
  3. S=\{ kd: k\in{\mathbb Z}^+\}.
  4. d is the least member of S.