{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Exam2Theorems

# Exam2Theorems - Math 302 Exam 2 Theorems Chapter 3 Rings...

This preview shows pages 1–2. Sign up to view the full content.

Math 302 Abstract Algebra Spring 2009 Exam 2 Theorems Chapter 3 Rings 3.2 BASIC PROPERTIES OF RINGS THEOREM 3.3 For any element a in a ring R , the equation a + x = 0 R has a unique solution. That is, for each a R , there exists a unique n R such that a + n = 0 R . THEOREM 3.4 (CANCELLATION PROPERTY) Let R be a ring. For all a, b, c R , if a + b = a + c , then b = c . THEOREM 3.5 For any elements a and b of a ring R , (1) a · 0 R = 0 R = 0 R · a . (2) a ( b ) = ( ab ) = ( a ) b . (3) ( a ) = a . (4) ( a + b ) = ( a ) + ( b ) . (5) ( a b ) = a + b . (6) ( a )( b ) = ab . If R has an identity, then for any a R (7) ( 1 R ) a = a . THEOREM 3.6 Let R be a ring and S be a nonempty subset of R that satisfies the following properties: SR1. Closure Under Subtraction. For all a, b S , a b S . SR2. Closure Under Multiplication. For all a, b S , ab S . Then S is a subring of R . THEOREM 3.9 Every field F is an integral domain. THEOREM 3.10 Cancellation is valid in any integral domain R . Suppose R is an integral domain. For all a, b, c R , a 6 = 0 R , if ab = ac , then b = c . THEOREM 3.11 Every finite integral domain is a field. 3.3 ISOMORPHISMS AND HOMOMORPHISMS THEOREM 3.12 Let f : R S be a ring homomorphism. Then (1) f (0 R ) = 0 S . (2) For all a R , f ( a ) = f ( a ) . (3) For all a, b R , f ( a b ) = f ( a ) f ( b ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Exam2Theorems - Math 302 Exam 2 Theorems Chapter 3 Rings...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online