Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring with unity.

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: 3. Let R be a commutative ring with unity. The intersection of all maximal ideals of R is called the…

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Q: Give an example of a subring of a ring, say A, that is not an ideal of A.

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Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…

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Q: Let R be a ring with unity 1 and char (R) = 3. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=3.  Then R contains a subring isomorphic to_______.

Q: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: Suppose A is a Noetherian unital commutative ring and I is an ideal of A. Prove that A/I is…

A: Some basic results of module theory required to understand the following proof better . If R is a…

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…

A: There are so many examples can be found.

Q: If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

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Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

Q: Let R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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Q: 3) Given a commutative ring with unity 1 in R; where R is a ring with two maximal ideals M₁ and Show…

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Q: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 20. Suppose R is a commutative ring and [R]= 30. If I is an ideal of R and |I| = 10, prove that I is…

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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: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: element a ∈ R, define the Annihilator of a denoted as Ann(a), as Ann(a) = {r ∈ R| r.a = 0} Show that…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an…

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Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.

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Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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Q: Let R be a finite commutative ring with unity.  Show that every prime ideal in R is a maximal ideal.

A: We have to show that if R be a finite commutative ring with unity then every prime ideal in R is a…

Q: Give an example (and justify your choice) of such a subring of a ring P[x] that contains P and is…

A: Given: Ring P[x] and a subring which contains P.

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

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Q: Let R be a commutative ring with an identity 1R and let J be a proper ideal with the property that…

A: Given R is a commutative ring with identity 1R                   J is a proper ideal of R R/J…

Q: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…

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Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

Q: Let f : R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is a…

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Q: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: 1. Let R be a commutative ring with unity and let a e Rbe fixed. Prove that the subset Ia = {x E R:…

A: i have provided the detailed proof in next step

Q: Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is…

A: Prime Ideal : A prime ideal S of a commutative ring R is a proper ideal of R such that a, b∈R and…

Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…

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Q: elements

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Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: Indicate such a subring of the ring P[x], which contains P and is different from P, but is not…

A: Image is attached with detailed solution.

Q: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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Q: An element a of a ring R is called nilpotent if an = 0 for some positive integer n. Show that the…

A: Given :            R is a commutative ring.            An element a∈R of a ring R is called…