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Algebraic Sturctures

Lesson #1

You will recall that a ring is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), provided that it is an ambelian group under addition.

Recall also that an ambelian group is a group that has the commutative property under the given binary operation, in this case -- addition.

In other words, for all a, b, c in the set:

1.  a + b = b + a

Moreover, addition is associative (by definition of a group).

2.  (a + b) + c = a + (b + c)

Also, addition has identity (by definition of a group).

3.  There is an element 0 and are such that a + 0 = a

Then of course, addition has an inverse (by definition of a group).

4.  There is an element – a in R such that a + (– a) = 0

Multiplication is also associative.

5.  a(bc) = (ab)c

And for all of the rings elements, the left distributive law and the right distributive law hold true.

6.  a(b + c) = ab + ac and (b + c)a = ba + ca

What is a Ring?

A B S R A C T   A L G E B R A

Note that multiplication need not be commutative for a group to be a ring, but if it is, we say that they ring is commutative.

Note also that a ring need not have an identity under multiplication, but if a ring other than {0} does have identity under multiplication, we say that the ring has unity (or identity).

And finally, a nonzero element of a commutative ring with unity need not have a multiplicative inverse, but when it does, we say that the nonzero element is a unit of the ring. For example, a is a unit if a¯¹ exists. (See pages 225 and 226 in Contemporary Abstract Algebra.)

A ring is the most general algebraic structure with two binary operations that we will study.

Again . . .

A ring is a set, together with two binary operations -- addition and multiplication -- defined on the set such that the ring constitutes an abelian group in which multiplication is associative and both the left distributive law and the right distributive law hold for all elements of the ring.

(See pages 253 and 254 in Abstract Algebra.)

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