(See Exercise 51.)
a. Write out the elements of
this ring. (Suggestion: Write
b. Is
c. Identify the unity elements, if one exists.
d. Find all units, if any exist.
e. Find all zero divisors, if any exist.
f. Find all idempotent elements, if any exist.
g. Find all nilpotent elements, if any exist.
Exercise 51.
Let
be arbitrary rings. In the Cartesian product
and
Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of
Prove that
Prove
has a unity element if both
have unity elements.
Given as example of rings
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Chapter 5 Solutions
Elements Of Modern Algebra
- 42. Let . a. Show that is a commutative subring of. b. Find the unity, if one exists. c. Describe the units in, if any.arrow_forward28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forward35. The addition table and part of the multiplication table for the ring are given in Figure . Use the distributive laws to complete the multiplication table. Figurearrow_forward
- Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a zero-divisor. We start the proof by (ab) e = 0, e# 0. Which of the following is a true statement in the proof? If ac = 0 then a = 0 If ac + 0 then b = 0 If ac = 0 then a is a zero divisor If bc = 0 then a = 0arrow_forwardCorrect answer asap.Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division ring is a field.arrow_forwardLet R = Z6 O Z4 and I = ((4, 2)). Write the elements of the factor ring R/I. Give the addition and multiplication tables for R/I.arrow_forward
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- 12. Let R² be the set of all pairs of real numbers R² = {(a, b)la, b = R}. The addition is still coordinate-wise but multiplication is defined by (a₁, b₁) * (a2, b₂) = (a₁a2b₁b₂, a₁b₂+ a2b1). The set R² equipped with these operations is a commutative ring. (You don't need to show this.) What is the unity element? Explain. What is the multiplicative inverse for (a, b) (0,0)? Explain. Find (x, y) such that (x, y) * (x, y) = (-1,0). Explain.arrow_forwardQ2: (A) Choose the correct answer for each of the following: 1. The cancellation law holds in a. any ring b. a commutative ring 2. The ring of even numbers (Ze, +, .) is a. with identity b. with zero divisor 3. The elements 2 and 4 have multiplicative inverses in the ring a. (Z6, +61-6) b. (Z7, +7,-7) c. (Q,+,.) 4. The ring is a field. a. (R- {0}, +..) b. (Z4, +44) 5. The set of odd integers is c. (R, +,.) with the usual addition and multiplication. c. ring without identity a. not ring b. ring with identity 6. Every ideal in the ring (Z, +,.) is a. principal b. maximal c. prime 7. Z10 is since 2 and 5 are zero divisors. a. an integral domain b. not an integral domain c. not ring c. an integral domain ring c. without identityarrow_forwardDetermine which elements of Z8 have inverses under the operation ×8. Then do the same for Z7 under ×7. Speculate on what is happening here.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage