3. A ring < R +, > are defined busing THREE (3) axioms that need to satisfy such as Abelian Group, Associative and Distributive Laws. Hence, show that is a ring.
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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here][Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .An element x in a multiplicative group G is called idempotent if x2=x. Prove that the identity element e is the only idempotent element in a group G. (Sec. 5.1, # 38) Sec. 5.1, # 38: 38. An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().
- 40. Let be idempotent in a ring with unity. Prove is also idempotent.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+y410. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
- A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.