In Exercises 1 through 6, compute the product in the given ring. (12)(16) in Z24 3. (11)(-4) in Zis 5. (2,3)(3,5) in Z, x Z, (16)(3) in Z32 6. (-3,5)(2,-4) in Z₁ x Z₁₁

Q: How can I prove that Z4 is cylic? I know that It has something to do with generators (I think?) and…

A: We have a group Z4. We have to prove that Z4 is cyclic.

Q: Let Z be the ring of integers, and let R be a ring without identity. Let S = Z x R be the ring with…

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Q: 6. Consider the ring Z₁2 with ideal I = {0, 3, 6, 9}. Give the ring tables of Z12/I.

A: 6. Consider the ring  with ideal .Give the ring tables of .

Q: . Consider the ring Z with ideal . Give the ring tables of Z/ .

A: Consider the ring  with ideal HereLet R = We know thatAndWe have to write the addition and…

Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

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Q: 4. Write out the addition and multiplication tables for each congruence class ring below, then…

A: Given that Congruence class ring  ℤ3x/x2+1 and ℤ2x/x2+1. We have to write addition and…

Q: 5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b)…

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Q: 3) Let Z, be ring integers mode 5 and f(x) € Zg(x) such that f(x)=x² + 2x + 1 then a) f(x) has only…

A: Since you have posted multiple questions, we will provide the solution only to the first question…

Q: 2. In this problem we will consider the units in the ring Z. (a) Show that 1 and 1 are both units in…

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Q: 15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and…

A: Subring Test- Let R be a ring. A non-empty subset S of R is is said to be a subring of R if S is…