(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
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- Below are two sets of real numbers. Exactly one of these sets is a ring, with the usual addition and multiplication operations for real numbers. Select the one which is a ring. O[a+bn:a, bez} O[a+b√2: a, b e Z} Let R be the ring above. True or false R is a ring with identity. OTrue OFalse R is a skewfield. OTrue OFalse R is a commutative ring. OTrue OFalsearrow_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_forwardWhat are the elements of 2Z/8Z? List down without spaces and in increasing coset representative. Hb. 1+8Z,2+8Z,... (on your solutions, give the addition and multiplication table of this quotient ring)arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage