Download Math 307 Abstract Algebra C.K. Li Notes on Chapters 12

Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 12-13
Chapter 12
Definition
A ring is a set with two binary operations: addition a ` b and multiplication ab such that
(R1) pR, `q is an Abelian group with identity 0, and inverse ´a for a P R.
(R2) pabqc “ apbcq for any a, b, c P R.
(R3) apb ` cq “ ab ` ac and pb ` cqa “ ba ` ca for any a, b, c P R.
Remark A ring may not have unity (multiplicative identity).
The multiplication may not be commutative. If it does, we say that R is commutative.
Examples Z, R, Zn , M2 pZq, Zrxs, external direct product R1 ‘R2 , the set of real-valued functions
f such that f p1q “ 0.
Theorem 12.1-2 Let a, b, c, be elements of a ring R. Then
1. a0 “ 0a “ 0.
[Proof. 0 ` 0a “ 0a “ p0 ` 0qa “ 0a ` 0a.]
2. ap´bq “ p´aqb “ ´pabq.
[Proof. ap´bq ` ab “ ap´b ` bq “ a0 “ 0 “ ´pabq ` ab.]
3. p´aqp´bq “ ab.
4. apb ´ cq “ ab ´ acq and pb ´ cqa “ ba ´ ca.
Suppose R has a unity 1.
5. The unity is unique.
6. p´1qa “ ´a
[Proof. 1 “ 111 “ 11 .]
[Proof. p´1qa “ ´p1aq “ ´a.]
7. p´1qp´1q “ 1.
8. Every a P R has none or a unique multiplicative inverse.
[Proof. If a1 a “ aa1 “ 1 and a´1 a “ aa´1 “ 1, then a´1 “ a´1 aa1 “ a1 .]
Definition A subset S of a ring R if pS, `, ¨q is a ring.
Theorem 12.3 A non-empty subset S of a ring R is a subring if and only if S is closed under
subtraction and multiplication, i.e., a ´ b, ab P S for any a, b P S.
Proof. If S is a subring, then a ´ b, ab P S for any a, b P S.
Suppose a ´ b, ab P S for any a, b P S. Then pS, `q is a group and ¨ is binary on S. For any
a, b, c P S, we can regard them as elements in R so that pabqc “ apbcq, apb ` cq “ ab ` ac, and
pb ` cqa “ ba ` ca.
Example Z Ď Q Ď R Ď C;
Zrxs Ď Qrxs Ď Rrxs Ď Crxs;
M2 pZq Ď M2 pQq Ď M2 pRq Ď M2 pCq;
Zris.
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Chapter 13 Integral Domains
Definitions
(a) A nonzero element a in a commutative ring R is a zero divisor if there is a nonzero element
b such that ab “ 0.
(b) An integral domain is a ring with unity and no zero-divisors.
(c) A field is a commutative ring R such that pR˚ , ¨q is a group.
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Examples Z, R, Zn , M2 pZq, Zrxs, Qr 2s, Z3 ris, external direct product R1 ‘ R2 .
Theorem 13.1 If a P R is not a zero divisor and ab “ ac, then b “ c. Consequently, if R is an
integral domain and a P R is nonzero such that ab “ ac, then b “ c.
Theorem 13.2 A finite integral domain is a field.
Corollary If p is a prime, then Zp is a field.
Definition If there is x P R such that nx “ x ` ¨ ¨ ¨ ` x ‰ 0 for any n P N, then we say that
R has has characteristic 0. Otherwise, we can let n be the smallest positive integer such that
0 “ nx “ x ` ¨ ¨ ¨ ` x (n times) for every x P R; we say that R has characteristic n. Let charR
denote the characteristic of R.
Theorem 13.3-4 If R has unity 1, then charR “ n if |1| “ n in pR, `q. If R is an integral domain,
then charR is zero or prime.
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