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Unformatted text preview: Math 302 Modern Algebra Spring 2009 Homework Solutions 3.2 BASIC PROPERTIES OF RINGS 3. Let R be a ring. (a) Prove that the additive identity (or zero element) is unique. Proof: Suppose 0 R and z R are both zero elements for the ring R . We need to show that z R = 0 R . Since 0 R is a zero element, then a + 0 R = a for all elements a R . In particular, applying this condition with a = z R , we obtain z R + 0 R = z R . Since z R is a zero element, then z R + a = a for all elements a R . In particular, applying this condition with a = 0 R , we obtain z R + 0 R = 0 R . Comparing these two equalities, we obtain z R = 0 R . Thus the zero element is unique. (b) Suppose R satisfies axiom R10. Prove that the multiplicative identity element is unique. Proof: Suppose 1 R and u R are both identity elements for the ring R . We need to show that u R = 1 R . Since 1 R is an identity element, then a 1 R = a for all elements a R . In particular, applying this condition with a = u R , we obtain u R 1 R = u R . Since u R is an identity element, then u R a = a for all elements a R . In particular, applying this condition with a = 1 R , we obtain u R 1 R = 1 R . Comparing these two equalities, we obtain u R = 1 R . Thus the identity element is unique. 5. Let R be a ring. (a) Prove: For all subrings S and T of R , S T is a subring of R . Proof: Suppose S and T are arbitrary subrings of R . We need to show S T is a subring of R . To show this, we can show that S T is nonempty and closed under subtraction and multiplication. Since S is a subring, then it contains the additive identity element 0 R of R . Similarly, since T is a subring, then it also contains 0 R . Therefore 0 R S T and so S T is nonempty....
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This note was uploaded on 10/20/2009 for the course MATH 302 taught by Professor Edwards during the Spring '03 term at CSU Fullerton.
 Spring '03
 Edwards
 Algebra

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