Page 41 - A Course in Linear Algebra with Applications

1.3: Matrices over Rings and Fields 25

The type of abstract algebraic system for which this can
be done is called a ring with identity. By this is meant a set
R, with a rule of addition and a rule of multiplication; thus
if ri and r 2 are elements of the set R, then there is a unique
sum r\ + r 2 and a unique product ir 2 in R- In addition the
r
following laws are required to hold:
(a) 7*1 + r 2 = r 2 + i, (commutative law of addition):
r
(b) (7*1 + r 2 ) + r 3 = ri + (r 2 + r 3 ), (associative law of
addition):
(c) R contains a zero element OR with the property
r + OR = r :
(d) each element r of R has a negative, that is, an
element —r of R with the property r + (—r) = 0.R :
(e) (rir 2 )r3 = ri(r 2r^), (associative law of
multiplication):
(f) R contains an identity element 1R, different from 0^,
such that r\R — r = #r :
l
r
(g) ( i + ^2)^3 = f]T3 + ^2^3, (distributive law):
r
(h) ri(r 2 + 7-3) = ir 2 + 7*17-3, (distributive law).
These laws are to hold for all elements 7*1, r 2 , r 3, r of the
ring .R . The list of rules ought to seem reasonable since all of
them are familiar laws of arithmetic.
If two further rules hold, then the ring is called a field:
(i) rxr 2 = r 2 r i , (commutative law of multiplication):
(j) each element r in R other than the zero element OK
has an inverse, that is, an element r" 1 in R such that
rr x = If? = r 1 r.
So the additional rules require that multiplication be a
commutative operation, and that each non-zero element of R
have an inverse. Thus a field is essentially an abstract system
in which one can add, multiply and divide, subject to the usual
laws of arithmetic.
Of course the most familiar examples of fields are