Show that the rings (3Z/60Z)/(12Z/60Z) and 3Z/12Z are isomorphic. Then show tha both isomorphic to Z4. are

Q: In commutative ring with identity each proper ideal is contained in prime ideal. O T OF

A:

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

A:

Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

Q: If the ring R has a left identity as Well right identity then these two as are equal.

A:

Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

A:

Q: If Ø: R S is a ring homomorphism. The Ø preserves:

A:

Q: Suppose that Φ: R --> S is a ring homomorphism and that the imageof Φ is not {0}. If R has a…

A:

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: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: ind subrings R1 and R2 of Z such that R1 union R2 is not a subring of Z

A:

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: is the ring 2Z isomorphic to the ring 4Z??

A: We can apply definition

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

A:

Q: Show that a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

A:

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: determine all ring homomorphism from Z20 to Z30

A:

Q: The number of nilpotent elements in the ring Z23 is: 1 4

A:

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 2- Let f be an isomorphism from the ring (R, +,) to the ring (R', +','). If (I, +;) is an ideal of…

A:

Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &6.

Q: 4. Give addition and multiplication tables for 2Z/8Z. Are 2Z/8Z and Z4 isomorphic rings? Concents

A: Let, nℤ be an ideal of a ring ℤ. Then the additive and multiplicative Cosetof nℤ can be defined as,…

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

A:

Q: Show that 2Z ∪ 3Z is not a subring of Z.

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: Show that the ring (2ot) is principal idenl ring.

A:

Q: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

Q: Find the charactes ist c of the ring Z2 Zg

A:

Q: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…

A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…

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: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

A:

Q: Is the mapping from Z5 to Z30 given by x --> 6x a ring homomorphism?Note that the image of the…

A: Given a mapping,

Q: IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p)…

A: Given N denote the set of non-invertible  elements of R.

Q: Z, show that the group 2Z/8Z is isomorphic to the group Z, but the ring 2Z/8Z is t isomorphic to the…

A:

Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…

Q: The rings Z and 5Z are isomorphic.

A:

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: Is the mapping from Z5 to Z30 given by x → 6x a ring homomorphism? Note that the image of the unity…

A: In the given question we have to tell that " Is f(x)=6x is a ring homomorphism from (Z5,⊕5,⊗5) to…

Q: It is known that 28= {0, 1, 2, 3, 4, 5, 6.73 is a Ring. H = { 0,43 is a subring of Z8. show that…

A:

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

A:

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

A:

Q: The ring Z,3 has exactly-------------maximal ideals

A:

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

A: Given: R is commutative ring

Q: is the ring 2Z isomorphic to the ring 3Z?

A: No ring 2Z is not isomorphic to the ring 3Z

Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

A:

Q: The number of nilpotent elements in the ring Z23 is:

A:

Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

A:

Q: Prove that the intersection of any collection of subrings of a ring Ris a subring of R.

A: Let S be  intersection of any collection of subrings of ring R. Then we have to prove  S is subring…

Q: The ring Z-[i] has no proper ideals

A:

Q: The ring Zp2, has exactly-----------maximal ideals

A: