Abstract Algebra-The Basic Graduate Year (2).pdf

page 1 of Chapter 2
CHAPTER 2 RING FUNDAMENTALS
Basic Definitions and Properties
Definitions ments A ring R is an abelian group with a multiplication
operation (a, b) → ab that is associative and satisfies the distributive laws: a(b+c)=ab+ac
and (a + b)c = ab + ac for all a, b, c ∈ R. We will always assume that R has at least two
elements,including a multiplicative identity 1 R satisfying a1R =1Ra = a for all a in R.
The multiplicative identity is often written simply as 1,and the additive identity as 0. If
a, b,and c are arbitrary elements of R,the following properties are derived quickly from the
definition of a ring; we sketch the technique in each case.
(1) a0=0a =0 [a0+a0=a(0+0)=a0; 0a +0a =(0+0)a =0a]
(2) (−a)b = a(−b)=−(ab)[0=0b =(a+(−a))b = ab+(−a)b,so ( −a)b = −(ab); similarly,
0=a0=a(b +(−b)) = ab + a(−b), so a(−b)=−(ab)]
(3) (−1)(−1) = 1 [take a =1,b= −1 in (2)]
(4) (−a)(−b)=ab [replace b by −b in (2)]
(5) a(b − c)=ab − ac [a(b +(−c)) = ab + a(−c)=ab +(−(ac)) = ab − ac]
(6) (a − b)c = ac − bc [(a +(−b))c = ac +(−b)c)=ac −(bc)=ac − bc]
(7) 1 = 0 [If 1 = 0 then for all a we have a = a1=a0 = 0,so R = {0},contradicting the
assumption that R has at least two elements]
(8) The multiplicative identity is unique [If 1
is another multiplicative identity then
1=11
=1
]
Definitions ments If a and b are nonzero but ab = 0,we say that a and
b are zero divisors;ifa ∈ R and for some b ∈ R we have ab = ba = 1,we say that a is a unit
or that a is invertible.
Note that ab need not equal ba; if this holds for all a, b ∈ R,we say that R is mutative
ring.
An integral domain is mutative ring with no zero divisors.
A division ring or skew field is a ring in which every nonzero element a has a multiplicative
inverse a−1
(., aa−1 = a−1a = 1). Thus the nonzero elements form a group under multiplication.
A field is mutative division ring. Intuitively,in a ring we can do addition,subtraction
and multiplication without leaving the set,w