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.

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Q: finite commutative ring with more than one element ar સા

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A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

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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…

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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.

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Q: 12. Let R² be the set of all pairs of real numbers R² = {(a,b)|a, b = R}. The addition is still…

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Q: Let R be a commutative ring with unity. Suppose that the only idealsof R are {0} and R. Show that R…

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Q: Let R be a ring with unity. Assume b1a=1R and ab2=1R. Prove that b1=b2.   Also, let R be a ring…

A: Ring with unity,so 1R is different from 0R ,

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Q: Let R be a commutative ring with unity. Fix two elements a, b ∈ R. Prove that if a = bt for some t ∈…

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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: 1) Let R be a commutative division ring. Then R[x] is a field. 2) Let R be a unique factorization…

A: We apply mod p test for 5th statement .

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

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Q: Zero Divisors: Let R be a ring, a, b, c ∈R, and fix m ∈Z. (a) Show that S = {r ∈R : mr = 0R } is a…

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Q: Let R be a ring and a, b in R. Show that ab is nilpotent implies that ba is nilpotent.

A: We know that An element "a" in a ring R is called nilpotent if an = 0 for some positive integer n.…

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

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

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

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

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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…

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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…

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Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: Prove: Let R be a ring with identity, and let x ∈ R. Then the equation (-1)x = -x is true in R

A: We use properties of Rings