1. Proof Problem (Rings): Let R be a ring. An idempotent of R is an element a e R such that a? is the multiplicative identity, 1 (if R is a ring with unity). Suppose R is a ring with unity and a E R is an idempotent element. Prove the following: = a. An example of an idempotent element (a) a(1 – a) is an idempotent element. (b) 1- a is an idempotent element. (c) 2a – 1 is invertible. That is, there exists an element x E R such that x(2a – 1) = 1. -

1. Proof Problem (Rings): Let R be a ring. An idempotent of R is an element a e R such that a? is the multiplicative identity, 1 (if R is a ring with unity). Suppose R is a ring with unity and a E R is an idempotent element. Prove the following: = a. An example of an idempotent element (a) a(1 – a) is an idempotent element. (b) 1- a is an idempotent element. (c) 2a – 1 is invertible. That is, there exists an element x E R such that x(2a – 1) = 1. -

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 4E: An element x in a multiplicative group G is called idempotent if x2=x. Prove that the identity...

See similar textbooks

Similar questions

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+y4
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 .
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. 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.
Let R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11 by proving the following statements. a. x(y)=(xy) b. (x)(y)=xy c. x(yz)=xyxz d. (xy)z=xzyz Theorem 5.11 Additive Inverses and Products For arbitrary x,y, and z in a ring R, the following equalities hold: (x)y=(xy) b. x(y)=(xy) (x)(y)=xy d. x(yz)=xyxz (xy)z=xzyz
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]
A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .
32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show 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().