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Something About Rings

A graduate of MaEd Math, an LPT. An academician and a published writer.

Something About Rings

Definition:

An algebraic structure ( R, + ) is a ring if R is a non-empty set and addition and multiplication are binary operations such that:

Addition:(R, +) is an abelian group which implies that it follows law on:

1) Associativity

2) Presence of an Identity

3) Presence of Inverses

4) Distributivity

5) Commutativity

A ring may not be commutative under multiplication and may not have a multiplicative inverse.

Subtraction/Division

Subtraction operation is considered as addition of the inverse of the number to be added and the minuend. Division operation is considered as multiplication of the first factor and the reciprocal of the second factor.

a - b is rewritten as a + - b

a * b is rewritten as a * 1/b

commutative ring:

a * b = b * a for all a, b elements of R

Example of Rings:

Number System:

1) All of Z (Integers)

2) Q (Rational Numbers)

3) R (Real Numbers)

4) C (Complex Numbers)

Polynomials with real coefficients form a commutative ring with identity under the usual addition/multiplication. This is denoted by R[x].

Modular Arithmetic

Binary arithmetic on {0, 1} gives a 2-element commutative ring with identity. Considering addition/multiplication mod n { 0, 1, ...n-1 }.

The set of Natural numbers is not a ring. For the inverse of N does not belong to the set of natural numbers. The set of natural numbers does not include negative integers. Natural numbers are set of counting numbers which start at 1.

The set of Whole numbers is not a ring since set of whole numbers start with 0 and additive inverse does not belong to the set of whole numbers or the set of whole numbers does not include the set of negative numbers.

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