Section 7.1
The structure ring
Multiplication turns Z, Q, R, C, Z[X], Q[X], R[X], C[X] into monoids, whereas addition defines a group structure on each of these sets. These two structures are combined in the notion of a ring:
Definition
A ring is a structure
(R,+,*,0,1) consisting of a set R for which (R,+,0) is a commutative group
and for which (R,*,1) is a monoid, in such a way that the following
laws hold for all x, y, z
R:
- x * (y + z) = x * y + x * z (left distributivity);
- (y + z) * x = y * x + z * x (right distributivity).
Notation and terminology:
- + is called the addition,
- * is called the multiplication (the symbol * is often omitted),
- 0 is the zero element,
- 1 is the unit element of the ring,
- the ring is called commutative if the multiplication is commutative.
Definition
A subring of a ring (R,+,*,0,1) is a subset S of R
containing 0 and 1 such that, whenever x, y S,
we have
x + y, -x and x * y S.
In other words, (S,+',0) and (S,*',1),
where *' and +' are the restrictions to
S × S, are again a group and a monoid, respectively.
In yet other words, S is a subring of R whenever S is closed under the operations defining the ring structure on R.