Decide whether each of the following sets
subring, give a reason why it is not. If it is a subring, determine if
is commutative and find
the unity, if one exists. For those that have a unity, which elements in
inverses in
a.
b.
c.
d.
e.
f.
g.
h.
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Elements Of Modern Algebra
- 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_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardEach of the following rules determines a mapping where is the field of real numbers. Decide in each case whether preserves addition, whether preserves multiplication, and whether is a homomorphism. a. b. b. d. e. f. Unless otherwise stated, and denote arbitrary rings throughout this set of exercises. In Exercises2-5, suppose and are isomorphic rings.arrow_forward
- a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forwardAn element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.arrow_forward
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardLabel each of the following statements as either true or false. Every subring of a ring R is an idea of R.arrow_forwardAssume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.arrow_forward
- True or false Label each of the following statements as either true or false. 7. For the quotient ring of by the ideal is .arrow_forward32. 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?arrow_forwardLet R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,