If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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Q: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: 1. Let I and J be ideals of a ring R. Prove that IJ is an ideal of R.

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Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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Q: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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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: Let K, I and J be ideals of ring R such that both I and J are subsets of K with I C J. Then show…

A: Given:- K,I and J be ideals of Ring R Such that I⊂J⊂K (i) Claim: K/I is subring of R/I ∀r.s∈R K⊂R…

Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

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

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Q: The ring Z is isomorphic to the ring 3Z O True False

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Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: Let R be a ring with unity, n a positive integer and a, b e R. Prove: If ab = ba, then (a + b)"…

A: Mathematical Induction  Let us consider a statement P(n) 1. Prove the given statement form n=1…

Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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Q: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: For a fixed element a of a ring R, prove that the set {x ϵ R I  ax = O} is a subring of R.

A: Given :            The ring R and a fixed element 'a' of R.   To prove that the set x ∈ R | ax =…

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Given that I = {(a,b) E R | a and b are even numbers} is an ideal of a ring R = Z × Z, construct…

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Q: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

A: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

Q: Let R be a ring. Prove that the set S = x R / xa = ax, a Ris a subring of R .

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: 30. Let R be a ring with identity lr and S a subring of R with identity 1s. Prove or disprove that…

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Q: Let M and N be ideals of a ring R and let H = {m+n | m∈ M, n ∈ N} (a) Show that H is an ideal of R.…

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Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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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 R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

Q: Let R be a ring with identity. If ab and a are units in R, prove that b is a unit.

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

Q: Prove that R[x2, x3] is a subring of the ring R[x].

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Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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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: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

Q: 3. Let R be a ring and b E R be a fixed element. Let and prove that T is a subring of R T = {rb | r…

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Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

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Q: Let R be a ring and S a subring of R. Prove that if 1R ∈S, then S has unity and 1S =1R

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Q: Find two elements a and b in a ring such that both a and b are zerodivisors,a + b ≠ 0, and a + b is…

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Q: If R is a commutative ring, shw that the characteristic of R[x] is the same as the characteristic of…

A: Given: R is commutative ring

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

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Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: b. Let R be a nontrivial ring such that, for each 0 + a E R there exists unique element x in R such…

A: Note: According to Bartleby guidelines; for more than one question asked, only the first one is to…

Q: Let A be an ideal of a ring R. i) If R is commutative then show that R/A is commutative ii) If R…

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Q: If I1 and I2  are two ideals of the ring R, prove that Ii n 11 ∩ I 2  is an ideal of R.

A: Given  I1 and I2  are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…