very submodule of a noetherian (artinian) mod s noetherian (artinian). Every homomorphic image of a noetherian (arti module is noetherian (artinian). et M be an R-module, and let N be an R-subm of M. Then M is noetherian (artinian) if and only oth N and M/N are noetherian (artinian).

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: Find all maximal ideals of Z_12:

A: Given ring is Z_12

Q: 5. Show that M:= {ECR: E is countable or E is countable} is a o-algebra.

A:

Q: 3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is…

A: Sol

Q: For two R-modules Mand N Show that Hom (M,N) any with the composition is not agroup?

A: Given that M and N are two R-modules. We have to show that HomM,N is not a group under composition.…

Q: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

Q: Suppose R is a commutative ring with 1_R. Let P be a prime ideal of R, I be the principal ideal…

A: Let's analyze each statement and determine whether it is true or false:(i) P = aP: This statement is…

Q: Show that the set of nonunits in Z8 is an ideal of Z8

A:

Q: Every homomorphic module is noetherian (artinian). image of a noetherian (artinian)

A: Every homomorphic image of a noetherian (artinian) module is noetherian (artinian).We have to…

Q: How to prove that: If M is a finitely generated R-module and {m1,....,mn} in M are elements of M…

A: How to prove that: If M is a finitely generated R-module and {m1,....,mn} in M are elements of M…

Q: If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

A:

Q: A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R,…

A: Since, you have posted a multiple question. But according to guidelines we will solve the first…

Q: Let Z be the ring of integers, and let R be a ring without identity. Let S = Z x R be the ring with…

A:

Q: The set H= {0,2,3}is a subring of (Z6, +6,.6) integer T F ring module 6.

A:

Q: Define the binary operations + and . On Z by a +b = a+b-1 and a.b= a+b- ab for all ab belongs to Z.…

A: We will use basic properties of ring homomorphism and integral domain.

Q: show that an ideal A is maximal iff A is

A:

Q: Theorem Let A a o – algebra. Let P:A (0, 1] so that P(2) = 1 and it is additive. Then it is…

A: Given that A be a σ-algebra. Let P:A→0,1 so that PΩ=1 and it is additive. To prove that given parts…

Q: Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

A: Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

Q: Q) Put True (T) or Fuls (F) for the following: (1)Every prime ideal I of ring R is maximal ideal.…

A:

Q: Let M be Noetherian left R-modu

A:

Q: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: GeoGebrä Task 1: This rectangle has been decomposed along its diagonal. Recompose the two pieces to…

A: A rectangle is a closed quadrilateral with four sides and four angles. All the four angles of the…

Q: 1. If S is a subring of R, then S[a] is a subring of R{x].

A:

Q: Let M be an R-module, and let N be an R-submodule  of M. Then M is noetherian (artinian) if and only…

A: Given:M is an R-module, and N is an R-submodule of M. To Prove:M is Noetherian (Artinian) if and…

Q: 3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is…

A: Given: px=m0+m1x+...+mrxrq(x)=n0+n1x+...+nsxs To find: Mx≅Rx⊗RM

Q: How to show that: If M is a finitely generated R-module and mM = M, then M = 0.

A: How to show that: If M is a finitely generated R-module and mM = M, then M = 0.

Q: rove that any unital irreducible R-module is cyclic.

A: Any unital irreducible R-module is cyclic

Q: Every submodule and every homomorphic image of a semi-simple module is semi-simpl

A: Every sub-module and every homomorphic image of a semi-simple module is semi-simple

Q: If R is a ring and S is a subring of R, then S is  an R-module (module over R).

A: R is ring and S is a subring of R  Claim : S is an R -module (module over R)

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

A: False

Q: 1+xr 1+x* a) b)

A:

Q: Suppose A is a o-algebra on R and f: R -00, +00] such that f¹(E) E A, prove that (-0, +oo]. If is…

A: Step 1:Step 2:Step 3:Step 4:

Q: 2. (i) Let R = Z, let M be the left R-module Z, and let X be the subset, X = {12, 11,9} CM. Show…

A: It is given that,  R = ℤ  and M be the left R-module of ℤ and  X = 12, 11, 9⊂M be any subset of M.…