(B) Give an example for a commutative ring with identity.
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- 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 ].A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]
- 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.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.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
- Prove that a finite ring R with unity and no zero divisors is a division ring.a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.