An algebraic system with two binary operations + and •,

is an abelian group. 
is a semigroup.  The operation • is distributive over + that is, for any a, b, c Î S, a • (b + c) = a • b + a • c and
(b + c) • a = b • a + c • a.
Note:
If
Similarly if
Familiar examples of rings are the set of integers, real numbers, rational numbers, and complex numbers under the operations
of addition and multiplication.
The additive identity is denoted by 0 and multiplication identity by 1.
If
3.5.2 Field
A commutative ring
multiplicative inverse in S is called a field.
Note:
The ring of integers is an integral domain, which is not a field.
The rings of real numbers and rational numbers are examples of fields.
Example 1:
The algebraic system
fixed integer n over the set of integers is a ring.
Note that
In fact,
Example 2:
Let S be a set and P(S) its power set. On P(S) we define operations + and • as follows
A + B = {xÎ S / xÎ A È B and xAÇ B}
A • B = AÈ B for all A, B Î P(S)
Then it is easy to verity that <P(S), +, •> is a ring called the ring of subsets of S.
A subset R Í S ,where
restricted to R.
Example:
The ring of even integers is a subring of the ring of integers.
Let
g(a + b) = g(a) Å g(b) and g(a • b) = g(a) ¤ g(b).
Note:
The first condition is a group homomorphism from
to
Property:
The distributive property is preserved by ring homomorphism.
Proof:
Let
For any a, b, c Î R,
g[a • (b + c)] = g(a) ¤ g(b + c)
= g(a) ¤ [g(b) Å g(c)]
= [g(a) ¤ g(b)] Å [g(a) ¤ g(c)]
= g(a • b + a • c)
which proves the property
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