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Consider the set
a. Construct addition and multiplication tables for
b. Observe that
is a commutative ring with unity
unity in
c. Is
d. Does
e. Which elements of
have multiplicative inverses?
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Chapter 5 Solutions
ELEMENTS OF MODERN ALGEBRA
- 35. 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_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardWrite out the addition and multiplication tables for 5.arrow_forward
- Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition and multiplication as defined in 18, consider the following questions. Is S a ring? If not, give a reason. Is S a commutative ring with unity? If a unity exists, compare the unity in S with the unity in 18. Is S a subring of 18? If not, give a reason. Does S have zero divisors? Which elements of S have multiplicative inverses?arrow_forwardThe addition table and part of the multiplication table for the ring R={ a,b,c } are given in Figure 5.1. Use the distributive laws to complete the multiplication table. Figure 5.1 +abcaabcbbcaccab abcaaaabaccaarrow_forward22. Define a new operation of addition in by and a new multiplication in by. a. Is a commutative ring with respect to these operations? b. Find the unity, if one exists.arrow_forward
- Given that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forwardProve that a finite ring R with unity and no zero divisors is a division ring.arrow_forward42. Let . a. Show that is a commutative subring of. b. Find the unity, if one exists. c. Describe the units in, if any.arrow_forward
- 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 ].arrow_forward11. 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 .arrow_forward12. 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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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