Let I and J be ideals of a ring R. Prove or disprove (by counterexample) that the following are necessarily ideals of R. (a) I+J = {x+ y | x E I and y E J} (b) InJ

Q: Let R be a ring and x be an indeterminate. Prove that the rings R[x]and R[x2] are ring-isomorphic.

A: Given That : Two rings R[x]  and R[x2]  To Prove: Two ringsR[x],  R[x2]  are ring-isomorphic

Q: Let x, y, and z be an element of a ring Prove that x(y-z) = xy-xz and (y-z)x=yx-zx

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Q: Let S = {a + bi : a, b e Z, b is even}. Then S is %3D not an ideal of Z[i] an ideal of Z[i]

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Q: To submit Let R be a ring. Define a function e : R[x] × R → R by the rule e(a„x" + ·..+a¡x+a0,r) =…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: 8. Let R denote the ring 2Z1/(1+ 3i). (a) Show that i- 3 € (1+3i) and that [i] = [3] in R. Use this…

A: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: If R is a ring and f(x) and g(x) are of degrees 3 and 4 then f(x)g(x) must have degree 7.

A: Given that R is a ring. deg{f(x)}=3 deg{g(x)}=4

Q: Let R be a ring, and let ICR be an ideal. Prove that R/I is commutative if and only if xy - yx € I…

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: Let R be a ring. (a) Show that R has exactly one zero, 0R. (b) If R also has an identity, 1R, show…

A: Given: R is a ring.We need to prove the following:(a) Show that R has exactly one zero, 0R. (b) If R…

Q: Let R be a ring. For n greater than or equal to 0, let In = {a, an element of R | 5na = 0}. Show…

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Q: Determine all the values of the parameter b € R, for which the polynomial f = x³y + bxy³2² belongs…

A: Given information:f(x,y,z)=x3y+bxy3z2Ideal: (x2+2y2,xz−y)To find: The suitable value of the…

Q: 6. Let I and J be ideals in the ring R. (a) Prove that InJ is an ideal in R. (b) Prove that I+J =…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Find constraint equations (if any) that b must satisfy in order for Ax = b to be consistent. 1 2 3…

A: Given system of is of the form Ax=b we have to find the constraints for b so that the system has…

Q: Let S = {a+ bi |a, b e Z , b is even}. Show that S is a subring of Z[i], but not an ideal of Z[i].…

A: Given- S=a+bi| a, b ∈ℤ , b is even .To show that S is a subring of ℤ[i] , but not an ideal of ℤ[i].…