Rings and fields

An algebraic system with two binary operations + and •, is called a ring if

1. is an abelian group.
2. is a semigroup.
3. 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 is commutative, then is called a commutative ring.

Similarly if is a monoid, then is called a ring with identity.


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 is a ring and for any a, b Î S such that a ¹ 0, b ¹ 0, a• b ¹ 0 then is a ring without zero divisors. A communicative ring with identity and without divisors of zero is called an integral domain.

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3.5.2 Field

A commutative ring which has more than one element such that every non-zero element of S has a
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 n, +_n, *_n> consisting of equivalence classes generated by the relation congruence modulo n for some
fixed integer n over the set of integers is a ring.



Note that 6, +₆, *₆> is not an integral domain since [3] * [2] = [0]. On the other hand 7, + ₇, * ₇> is an integral domain.
In fact, n, +_n, *_n> is a field if and only if n is a prime integer.


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 is a ring, is called a subring if is itself a ring with operations + and •
restricted to R.

Example:

The ring of even integers is a subring of the ring of integers.



Let •> and Å , ¤ > be rings. A mapping of g : R® S is called a ring homomorphism from and
Å ,¤ > if for any a, b, Î R

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 Å > and second is a semigroup homomorphism form
to ¤ >.


Property:

The distributive property is preserved by ring homomorphism.

Proof:

Let and Å , ¤ > be rings and g : R® S be a ring homomorphism.

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