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

Q: 37. An element x in a ring is called an idempotent if x2 = x. Prove that the only idempotents in an…

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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: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…

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Q: Let R be a ring with unity in which r^2 = r for every r in R. Prover that if a does not equal 0 and…

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Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

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: 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: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: If a is unit in a ring R then a is Select one: a. invertable b. the additive identity C. zerodivisor…

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Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

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

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Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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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: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

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 I be an ideal of a commutative ring (R,+.)with unity, then I is no prime ideal iff RM is…

A: We know characterstic of an integral domain is either prime or zero. Let I be an ideal of a…

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: Let R be a commutative division ring. Then prove or disprove that R[x] is a field.

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Q: Let R be a commutative ring with unity. Fix two elements a, b ∈ R. Prove that if a = bt for some t ∈…

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Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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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 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: For any element a in a ring R, define (a) to be the smallest ideal of R that contains a. If R is a…

A: An ideal is a non-empty sub set I of a ring R, such that

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 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: Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.

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

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Q: Prove that a nonzero commutative ring with unity R is a field if and only if it has two ideals (0)…

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Q: 26. a. For a fixed element a of a commutative ring R, prove that the set I = {ar|rER} is an ideal of…

A: The objective is to prove that I is an ideal of R.

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: Prove that if u is a unit in a ring R, then -u is a unit in R.

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Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R = {a+…

A: a) Let F be a field. We know that field has no proper ideals. The ideals of F are 0 and F only. The…

Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

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: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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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: 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: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

A: Given :          ab is a zero divisor where a and b are elements in ring R. Formula :        Zero…

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 more than one element. Provethat if for every nonzero element a of…

A: We have given commutative ring R with more than one element. We have to prove R has a unity and…

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

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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 with unity  If I is a prime ideal of R, prove that I[x] is a prime…

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Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

A: We are given that R be a commutative ring with unity. We have to show that  R is a field if and only…

Q: Prove that the set of all elements in a ring R that are not zero divisors is closed under…

A: We will prove that the set of all elements in a ring R that are not zero divisors is closed under…