Show if R is a commutative ring then R[x] is also a commutative ring.
Q: Construct a homomorphism of rings p:Z[i] → Z,
A: Consider the rings ℤi and ℤ2. Define a map φ:ℤi→ℤ by φa+ib=0 ∀ a,b∈ℤ. Let a+ib, c+id∈ℤi.…
Q: Let R = {2n: n E Z} and define addition O and multiplication O in R by a.b a O b : = a + b and aOb…
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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: Give an example of a subring of a ring, say A, that is not an ideal of A.
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Q: Let K be a commutative ring. For every AÃ ÃA = det (A) In- = quence, A is invertible iff det(A) is…
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Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…
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Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.
A: Given : R is a ring with n elements. To prove : The characteristic of a ring R divides the number…
Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.
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Q: The ring Z is isomorphic to the ring 3Z
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Q: Find an example of a commutative ring A that contains a subset, say S, such that for every s E S we…
A: We can find n commutative ring A such that A contains a subset S such that for every s in S we have…
Q: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules
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Q: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…
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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: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…
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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: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O
A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…
Q: 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…
A: Since we have given that R is a ring with unity and N is a proper ideal of R . We prove R/N is a…
Q: The ring Z is isomorphic to the ring 3Z O True False
A: Solution:
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: 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: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…
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Q: If A and B are ideals of a commutative ring R, define the sum of A and B as: A+B = {a+b| a € A, b €…
A: Let A and B be two ideals of a ring R. If A and B are ideals of a commutative ring of R with unity…
Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…
A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.
Q: Let CR,t,,) be a Commutative ring ?
A: First I recall you Definition of commutative ring. A commutative ring is ring in which…
Q: Let R be a commutative ring with identity and let I be a proper ideal of R. Prove that R/I is a…
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Q: Let R be a ring with a subring S: Prove or disprove: If a ∈ R is a unit, and a ∈ S, then a is also a…
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Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…
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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. (b) If R is a…
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Q: Let R and S be isomorphism rings. (a) Prove that R ring with unity if and only if S is a ring with…
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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: Show that if n is an integer and a is an element from a ring, thenn . (-a) = -(n . a).
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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca, then b=c.
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Q: Provide an example of the following if possible. (a) An infinite ring R such that Char(R) = 97. (b)…
A: Note : for option (c) Take R=Q is simple ring and Q has infinitely many no simple subring…
Q: 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…
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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…
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Q: Let A, B,C be rings. Let o be a ring homomorphism from A into B and B be a ring homomorphism from B…
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Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring * O True O…
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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…
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Q: Prove that a finite ring R with unity and no zero divisors is a division ring.
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Q: Let R be a commutative ring with unity and r ∈ R. Prove that if ⟨r⟩ = R, then r is a unit. Consider…
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Q: Let R be a ring, and a, b e R.What is the needed condition on R such that the expression (a + b) (a…
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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 R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.
A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…
Q: Let R be a ring and M be an R-module. Ther
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Q: An element x in a ring is called an idempotent if x2 = x. Prove that the characterstic of R is 0 or…
A: An element x in a ring is called an idempotent if x^2 = x
Q: Let R be a ring such that a6 - = x for all æ E R. Prove that R is commutative.
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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…
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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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .19. Find a specific example of two elements and in a ring such that and .24. 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 .)
- 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.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.