Q: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.
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Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.
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Q: If the ring R has a left identity as Well right identity then these two as are equal.
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Q: If Ø: R S is a ring homomorphism. The Ø preserves:
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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: 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: 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: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…
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Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.
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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.
A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…
Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism
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Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).
A: Solve the following
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: If in a ring R every x E R satisfies x2 = x, Prove that R must be commutative.
A: Answer and explanation is given below...
Q: Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…
A: The statement is true.
Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite
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Q: Let R be a ring with 1. Show that R[z]/ (x) ~ R.
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Q: If u is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(EU F)+µ(EnF) %3D
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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 R be a ring with m elements. Show that the characteristic of Rdivides m.
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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, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…
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Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.
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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 commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and…
A: We shall solve first question only as you have asked more than one different question as per company…
Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.
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Q: (17) Prove that the ring Zm Xx Z, is not isomorphic to Zmn if m and n are not relatively prime.
A: We have to prove given property:
Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D
A: The term radical is used when we think about ideals and when we talk about ideals definitely…
Q: Let R be a commutative unitary ring and let M be an R-module. For every rERlet rM = {rx; x E M} and…
A: The complete solutions are given below
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: Prove that if u is a unit in a ring R, then -u is a unit in R.
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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0
A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…
Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…
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Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.
A: Solution
Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…
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Q: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…
A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…
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: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…
A: Take R = Z[i√5] Clearly, R is commutative ring and Z ⊆ R Also, 2,3 ∈ R are not prime but…
Q: Le 'R' be a ring with 1.If 'a' is milpotent ele ment of 'R', Prove thał 1-a and 1 ta are Units.
A: Ring: A ring R is a set with two binary operations, addition (denoted by a+b) and multiplication…
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: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).
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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: Theorem 12. Let R is a commutative ring and r e R,fe Hom (M, M'), then rfe. R M, M') defined by (rf)…
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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…
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- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y419. Find a specific example of two elements and in a ring such that and .
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.
- [Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]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.a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].