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_forwardWrite out the addition and multiplication tables for 5.arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.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_forwardThe number of zero divisors of the ring Z4 + Z5 isarrow_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