n this exercise we want you to find all ring homomorphisms g from Z/36Z to Z/42Z. List all the possible values of g(1) when g is a ring homomorphism from Z/36Z to Z/42Z. The possible values are: Notation: An element à € Z/kZ should be written as a number between 0 and k - 1. Enter the answer as a set of numbers, for example: {0, 1,4}.

Q: Q6: Define the Boolean ring. Is (Z,+,.) Boolean?

A: .

Q: 6. Prove or disprove: the set of all subsets of R is a ring with respect to the operations A…

A:

Q: Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a…

A:

Q: Abstract Algebra. Please explain everything in detail.

A: To describe all the field automorphisms of the given field.

Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…

A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

Q: Enter the multiplicative inverse of the element [10] x + [13] in the ring z-[x]

A: We need to fin the multiplicative inverse of the element [10]20x+[13]20 in the ring Z20x. Let…

Q: Let R be a ring with 10. A nonzero element a is called a left zero divisor in R if there is a…

A:

Q: Question: Let G1 = ℤ3 , G2 = ℤ4, and G3 = ℤ2 1. Find an isomorphism from G1⨁G3 to ℤ6

A:

Q: Demonstrate or explain why the system (P(R#), +, *) is a ring with identity and is a commutative…

A: (i) Let,           f(x) = anxn + an-1xn-1 + . . . . . . . . + a1x + a0   ∈ P(R#)        Then,…