6 RINGS, INTEGRAL DOMAINS, AND FIELDS
Let R be a nonempty set with two binary operations, an operation of addition (denoted by +) and an operation of multiplication (denoted by juxtaposition). Then R is called a ring if the following axioms are satis?ed:
[R1 ] For any a, b, c ? R, we have (a + b) + c = a + (b + c).
[R2 ] There exists an element 0 ? R, called the zero element, such that, for every a ? R,
a + 0 = 0 + a = a.
[R3 ] For each a ? R there exists an element ?a ? R, called the negative of a, such that
a + (?a) = (?a) + a = 0.
[R4 ] For any a, b ? R, we have a + b = b + a. [R5 ] For any a, b, c ? R, we have (ab)c = a(bc).
[R6 ] For any a, b, c ? R, we have: (i) a(b + c) = ab + ac, and (ii) (b + c)a = ba + ca.
Observe that the axioms [R1 ] through [R4 ] may be summarized by saying that R is an abelian group under addition.
Subtraction is de?ned in R by a ? b = a + (?b).
One can prove (Problem 21) that a · 0 = 0 · a = 0 for every a ? R.
A subset S of R is a subring of R if S itself is a ring under the operations in R. We note that S is a subring of
R if: (i) 0 ? S, and (ii) for any a, b ? S, we have a ? b ? S and ab ? S.
Special Kinds of Rings: Integral Domains and Fields
This subsection de?nes a number of different kinds of rings, including integral domains and ?elds.
R is called a commutative ring if ab = ba for every a, b ? R.
R is called a ring with an identity element 1 if the element 1 has the property that a · 1 = 1 · a = a for every
element a ? R. In such a case, an element a ? R is called a unit if a has a multiplicative inverse, that is, an
element a?1 in R such that a · a?1 = a?1 · a = 1.
R is called a ring with zero divisors if there exist nonzero elements a, b ? R such that ab = 0. In such a
case, a and b are called zero divisors.
De?nition 3: A commutative ring R is an integral domain if R has no zero divisors, that is, if ab = 0 implies
a = 0 or b = 0.
De?nition 4: A commutative ring R with an identity element 1 (not equal to 0) is a ?eld if every nonzero
a ? R is a unit, that is, has a multiplicative inverse.
A ?eld is necessarily an integral domain; for if ab = 0 and a = 0, then
b = 1 · b = a?1 ab = a?1 · 0 = 0
We remark that a ?eld may also be viewed as a commutative ring in which the nonzero elements form a group under multiplication.
EXAMPLE 13
(a) The set Z of integers with the usual operations of addition and multiplication is the classical example of an integral domain (with an identity element). The units in Z are only 1 and ?1, that is, no other element in Z
has a multiplicative inverse.
(b) The set Zm = {0, 1, 2,...,m ? 1} under the operation of addition and multiplication modulo m is a ring; it is called the ring of integers modulo m. If m is a prime, then Zm is a ?eld. On the other hand, if m is not a prime then Zm has zero divisors. For instance, in the ring Z6 ,
2 · 3 = 0 but 2 ? 0 (mod 6) and 3 ? 0 (mod 6)
(c) The rational numbers Q and the real numbers R each form a ?eld with respect to the usual operations of addition and multiplication.
(d) Let M denote the set of 2 ×2 matrices with integer or real entries. Then M is a noncommutative ring with zero divisors under the operations of matrix addition and matrix multiplication. M does have an identity element, the identity matrix.
(e) Let R be any ring. Then the set R[x] of all polynomials over R is a ring with respect to the usual operations of addition and multiplication of polynomials. Moreover, if R is an integral domain then R[x] is also an integral
domain.
Ideals
A subset J of a ring R is called an ideal in R if the following three properties hold: (i) 0 ? J .
(ii) For any a, b ? J , we have a ? b ? J .
(iii) For any r ? R and a ? J , we have ra, ar ? J .
Note ?rst that J is a subring of R. Also, J is a subgroup (necessarily normal) of the additive group of R. Thus we can form the following collection of cosets which form a partition of R:
{a + J | a ? R}
The importance of ideals comes from the following theorem which is analogous to Theorem 7 for normal subgroups.
Theorem 10: Let J be an ideal in a ring R. Then the cosets {a + J | a ? R} form a ring under the coset operations
(a + J) + (b + J) = a + b + J and (a + J )(b + J) = ab + J
This ring is denoted by R/J and is called the quotient ring.
Now let R be a commutative ring with an identity element 1. For any a ? R, the following set is an ideal:
(a) = {ra | r ? R}= aR
It is called the principal ideal generated by a. If every ideal in R is a principal ideal, then R is called a principal ideal ring. In particular, if R is also an integral domain, then R is called a principal ideal domain (PID).
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .