practice_midterm_1_problems

practice_midterm_1_problems - if a 2 = a . Prove that if R...

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College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 1 Prof. Jones 1. Let I be an ideal in a ring R , and set N ( I ) = { x R : xi = 0 for all i I } . a) Show that N ( I ) is an ideal in R . b) Show that if R is an integral domain, then either I = { 0 } or N ( I ) = { 0 } . c) Give an example of a ring R and an ideal I 6 = { 0 } in R such that N ( I ) = R . 2. Let φ : R R 0 be a ring homomorphism, and let I be an ideal of R . a) Show that if φ is onto, then φ ( I ) is an ideal of R 0 . b) Give an example to show that the conclusion of a) may fail if φ is not onto. 3. Let R be a ring with unity, and suppose that R has 18 elements. Show that R cannot be an integral domain. 4. Recall that an element a of a ring R is an idempotent
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Unformatted text preview: if a 2 = a . Prove that if R is commutative and has characteristic 2, then the idempotent elements form a subring. 5. If R is a nite commutative ring with unity, prove that every maximal ideal of R is a prime ideal of R . 6. Let R be a commutative ring with unity that has the property that a 2 = a for all a R . Let I be a prime ideal in R . Show that the order of R/I must be 2. 7. Show that Z 3 [ x ] / h x 2 +1 i is ring-isomorphic to Z 3 [ i ] = { a + bi | a,b Z 3 } . Is the same statement true if we replace Z 3 with Z 7 ? 1...
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This document was uploaded on 11/03/2010.

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