Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].
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A: Let R be a ring with unity 1 and char(R)=3. Then R contains a subring isomorphic to_______.
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A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________
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A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0
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Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…
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Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).
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A: That's easy. Thumb up. Have a great day!!!
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A: Image is attached with detailed solution.
Q: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.
A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.
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isomorphic rings. Prove that R[r] = S[r].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5958936e-64d8-4f9f-ac16-af104f1eb32d%2Fea5f4f96-e9c2-470d-aef3-8628ce828d11%2Feeenyxn_processed.png&w=3840&q=75)
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- 19. Find a specific example of two elements and in a ring such that and .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.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.
- Let 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.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of24. 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 .)