A
ring R (with 1) is a
set equipped with two
binary operations usually denoted by
addition (+) and
multiplication
and an element
1 in
R such that:
- (R,+) is an abelian group
- a(bc)=(ab)c for all a,b,c in R (associativity)
- a(b+c)=ab+ac and (b+c)a=ba+ca for all a,b,c in R (distributivity)
- 1a=a1=a for all a in R (identity)
An example of of a commutative ring is the ring of integers Z={...,-1,0,1,2,...}
with the usual addition and multiplication.
An example of a noncommutative ring is the ring of all
nxn matrices with complex entries.
Mathematicians who contributed to ring theory include
David Hilbert and Emmy Noether
Rings turn up naturally in lots of places in mathematical
physics.
The Weyl algebra is an interesting noncommutative ring.