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
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardConsider 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_forwardLabel each of the following statements as either true or false. Every subring of a ring R is an idea of R.arrow_forward
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- Assume 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_forward15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forward12. Let be a commutative ring with unity. If prove that is an ideal of.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,