If Ø: R → S is a ring homomorphism. The Ø preserves: Units All of these Nilpotent elements Idempotent elements

Q: IF J is nil left ideal in an Artinian ring R, then J is

A:

Q: If fis a ring homomorphism from Zm to Zn such that f (1) = b, then b4k+2 = bk True False

A:

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

A:

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

A:

Q: If Ø: R → S is a ring homomorphism. The Ø preserves: Nilpotent elements O Idempotent elements O…

A:

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 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: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

A:

Q: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules

A:

Q: If fis a ring homomorphism from Zm to Z, such that f(1)= b, then b**2 = b*. True False

A:

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

A:

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

A:

Q: Let R = Q[V2] and S = Q[V3]. Show that the only ring homomorphism from R to S is the trivial one. In…

A:

Q: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: Ring homorphism

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: If Ø: R→ S is a ring isomorphism. The Ø preserves: O All of these O Nilpotent elements O Idempotent…

A:

Q: Let R and S be rings and let ø : R → S be a ring homomorphism. Show that o is one-to-one if and only…

A:

Q: Let f:R, » R, and let g : R, » R, be two ring isomorphisms. If ø =g•f, then 2- ø(e,) = a) , b) € c)…

A:

Q: If Ø: R → S is a ring isomorphism. The Ø preserves: O Nilpotent elements Units Idempotent elements O…

A:

Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

A:

Q: Let R and S be commutative rings. Prove that (a, b) is a zero-divisorin R ⨁ S if and only if a or b…

A:

Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: Suppose that : R → S is a ring isomorphism. Then g-l is a ring homo- morphism, and hence a ring…

A: According to the given information:

Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

A:

Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

A:

Q: If A is a unitary ring define its centre to be CenA = {x €A; Vy EA) xy = yx}. %3D Show that CenA is…

A: Unitary ring is s ring with unity ie  A ring containing multiplicative identity.

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

A:

Q: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: Determine whether the following mapping is a ring homomorphism h: Z[V2]→ Z[v2] defined by h(a +…

A:

Q: Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…

A: Given that R is a commutative ring with identity.

Q: If Ø: R → S is a ring homomorphism. The Ø p O Idempotent elements Units All of these Nilpotent…

A: For a ring R, the element a∈R is idempotent if a2=a For a ring R, the element a∈R is nilpotent if…

Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: Suppose phi maps R to S is a ring homomorphism and the image of of phi is not {0}. If R has a unity…

A:

Q: If fis a ring homomorphism from Zm to Z, such that f (1) = b, then b*+2 = b*. True False

A:

Q: 6. Let R be a commutative ring with identity and let f be an onto homomorphism from R to a ring S.…

A:

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: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the covering and M⊆R is maximal ideal, then…

A: Given φ:R→S be a ring homomorphism. Let,, φ is covering and M⊂R is maximal ideal.  To prove that…

Q: 1. let S: :1 xg*Z a) Show that is a ning. ning. b) Show that ¢: S Z olefined by $ ( [Š 87)-* is a…

A: Solution of part(a): It is given that S=x0y0, x,y∈ℤ. We know that the sum and product of two 2×2…

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…

A:

Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

A:

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…

A:

Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

A:

Q: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

A:

Q: f:(R,+) (R', +'/) be a ring Homomorphism, and R is integral domain, then R' is integral domain if f…

A:

Q: Iff is a ring homomorphism from Zm to Zn such that f (1)=b, then b*+2 = b*. False True

A:

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

A:

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

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.