Abstract Algebra by Lloyd R. Jaisingh

Author:Lloyd R. Jaisingh [Jaisingh, Lloyd R.]
Language: eng
Format: epub
Publisher: McGraw-Hill
Published: 2004-01-09T05:00:00+00:00

11.4 TYPES OF RINGS

DEFINITION 11.4: A ring for which multiplication is commutative is called a commutative ring.

EXAMPLE 6. The rings of Examples 1, 2, 3(a) are commutative; the ring of Example 3(b) is non-commutative, i.e., b · c = a, but c · b = c.

DEFINITION 11.5: A ring having a multiplicative identity element (unit element or unity) is called a ring with identity element or ring with unity.

EXAMPLE 7. For each of the rings of Examples 1 and 2, the unity is 1. The unity of the ring of Example 3(a) is b; the ring of Example 3(b) has no unity.

Let ℛ be a ring of unity u. Then u is its own multiplicative inverse (u−1 = u), but other non-zero elements of ℛ may or may not have multiplicative inverses. Multiplicative inverses, when they exist, are always unique.

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