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- 19. Find a specific example of two elements and in a ring such that and .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?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.
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.32. Consider the set . a. Construct addition and multiplication tables for, using the operations as defined in . b. Observe that is a commutative ring with unity, and compare this unity with the unity in . c. Is a subring of ? If not, give a reason. d. Does have zero divisors? e. Which elements of have multiplicative inverses?15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
- 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 abcaaaabacca22. 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.Let a0 in the ring of integers . Find b such that ab but (a)=(b).
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.Exercises If and are two ideals of the ring , prove that is an ideal of .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. Figure