Consider the ring R= [r.s.t) whose addition and multiplications ubles are given below, Then t.s= S
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A: Ans : 4 Option 2nd true
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Q: Consider the ring R = {r, s,t} whose addition and multiplications tables are given below. + |r rrr…
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Q: Consider the ring R = {r,s,t} whose addition and multiplications tables are given below. Then t.s =
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either S,…
A: I jave used the definition of subring
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Q: a) Let R be a ring Ei a3 = a #aER %3D Prove that R is commutatve.
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Q: Consider the ring R = [r, s,t) whose addition and multiplications tables are given below, Then ts S…
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- Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)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.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of
- 14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.Let a0 in the ring of integers . Find b such that ab but (a)=(b).An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.