If the statement is absolutely true, write Always True; if it is true for some situation, write Sometimes; but if it is never true, write Never. 1) Let R be a commutative ring. Then R has must contain a unity. 2) Let R be a ring with unity whose characteristic is 6. R is a field. 3) Let R be a finite commutative ring with unity 1≠0 in which both left and right cancellation laws hold. R must be a field. 4) Let R be a principal ideal domain. For two elements a and b in R, a is an associate of b iff (a)=(b). 5) Let S be a subdomain of R (S≤R). If f is irreducible over R then f is irreducible over S.

Q: If fis a ringhomomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: given a ring homomorphism.

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

A:

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: 7) Let R be a ring with 1, M a left R – Module, and N a submodule of M. Prove that if both M/N and N…

A:

Q: cg. 2. Let a, b, c, f, and g be elements of a commutative ring R with a‡OR. Prove that if a divide…

A:

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

A:

Q: 12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and…

A: Given that R is a ring without zero divisors. If ab=1, show that ba=1

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

A:

Q: Let R be a commutative ring with identity, and let a, b ∈ R. If a² + b² = 0, prove that (a + b)² =…

A: In algebra, a ring is a set equipped with two binary operations - addition and multiplication, that…

Q: 3. Let R be a commutative ring. Let x, y E R such that their characteristics are p. Show that (x +…

A:

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

A:

Q: Let Rbe a ring, and a, b e R.What is the needed condition on R such that the expression (a + b) (a –…

A: Given that ( a + b )( a - b ) = a^2 - b^2                 Or,  a^2 - ab + ba - b^2 = a^2 - b^2…

Q: 3. Consider the set {a+b√2: a, b E Z} with the usual addition and multiplication. Show that this is…

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: Give an example of ring elements a and b with the properties thatab = 0 but ba ≠ 0.

A: We have to give the example of ring elements that follow the following properties: ab=0 but ba≠0.

Q: J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1…

A:

Q: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

A:

Q: A ring may have more than one identity element for multiplication? T or F

A:

Q: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

Q: Either prove or give a counterexample: If R is a ring with zero divisors, then R[x] is a ring with…

A: (A) Given that R is a ring with zero divisors.

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: 21. Let R be a ring and let a be a nonzero element of R that is not a zero divisor. Prove that…

A: 21. Let R be a ring and let a be a nonzero element of R that is not a zero divisor.   To Prove:…

Q: True or False: Any ring must be commutative with identity.

A:

Q: 1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes…

A:

Q: Which one of the following is true? In the ring Z6, 4 does not divide 2 In the ring Z6, if x² = x…

A: Notice the some points regarding ℤn : ℤn=0,1,2,3,⋯⋯⋯⋯⋯,n-1. If x,y∈ℤn and x divides y then there…

Q: Show, by example, that for a commutative ring R, it is possible that deg(fg) < deg f + deg g where…

A: Given: fx and gx are non-zero elements in Rx. To prove: deg fxgx&lt;deg fx+deg gx by an example.

Q: Q[V5 is Principal Ideal Domain. True O False

A: Since you have asked multiple questions, as per our company guidelines, we will solve first question…

Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

A: