(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
- The 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_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_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_forward
- Abstract Algebra:arrow_forwardLet R be a ring. The multiplication table has several entries missing as shown in attached image. Using only the ring axioms, complete the table.arrow_forwardLet 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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage