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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

5. *a*(*bc*) = (*ab*)c

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

What is a Ring?

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 . . .

Memorize it!: