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- Prove that a finite ring R with unity and no zero divisors is a division ring.22. Let be a ring with finite number of elements. Show that the characteristic of divides .33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)
- An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.
- a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].True or False Label each of the following statements as either true or false. If one element in a ring R has a multiplicative inverse, then all elements in R must have multiplicative inverses.40. Let be idempotent in a ring with unity. Prove is also idempotent.
- 46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.a. Give an example where a and b are not zero divisors in a ring R, but the sum a+b is a zero divisor. Give an example where a and b are zero divisors in a ring R with a+b0, and a+b is not a zero divisor. Prove that the set of all elements in a ring R that are not zero divisors is closed under multiplication.True or False Label each of the following statements as either true or false. 8. A unity exists in any commutative ring.