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 unit in S.

Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

Q: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

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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: As a ring, z is isormorphic to nZ for all n>1. True False

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Q: Let R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…

A: we will use sub ring test.

Q: Let R be a ring of all real numbers , Show that H= { m+nV2 | m, ň E Z} is a subring of R.

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Q: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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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: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).

A: Solve the following

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

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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: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

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: 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: 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 S be a ring. Determine whether S is commutative if it has the following property: whenever ry =…

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Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

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: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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Q: A ring, R, in which r^2 = r for all elements r in R is called a Boolean ring. In any Boolean ring,…

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Q: Let R be a ring and a an element in R. Set Ia = {x∈ R so that x ⋅ a = 0}. Show that Ia is a sub ring…

A: given Ia=x∈R so that x.a=0to prove Ia is subring (i) x1 and x2∈Ia such that   x1.a=0 , x2.a=0  ⇒(…

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

A: Z=···,-3,-2,-1,0,1,2,3,···3Z=···,-9,-6,-3,0,3,6,9,··· As the ring Z has the unity element 1 such…

Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Let A be a ring with identity e and let a, b E A such that ab = e. Then ba is a Idempotent element…

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Q: The center of a ring R is {z ∈ R : zr = rz for all r ∈ R}, i.e. the set of all elements which…

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

A: According to the given information, It is required to prove that if u is a unit in a ring R then -u…

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: 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: 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: Suppose that R is a ring and that a2 = a for all a in R. Show that Ris commutative. [A ring in which…

A: Given: R is a ring such that a2=a for all a in R. To show: R is commutative ring.

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: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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Q: (d) Let S = (Q, +, ·), the ring where + is addition of rational numbers and · is multiplicaiton of…

A: The given question is related with abstract algebra. We have to solve the following :…

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.