If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an example to show that this may fail if the ring is not commutative( maybe using a counterexample (a+b)6=0)

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

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Q: Show that the set Q(√2) = {a+b√2, a, b € Q} is a ring, under operations of multiplication and…

A: The set Q2=a+b2 , a , b∈Q  and operations under multiplication and addition define by :…

Q: (a) Let R be a ring, a € R and define AR = {r € R|ar = ar = OR} where OR ER is the zero element of…

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Q: Decide whether z(sqrt(-5)) unique factorization domain or not (ring theory)

A: To decide whether Z(√-5) is unique factorization domain or not.

Q: 29 justifying each step you use for this. 57 b) Calculate

A: (b) The objective is to find the value for 2957. By using the division algorithm, 29=0×57+29. So,…

Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Can someone please help me understand the following problem. I need to know how to start the…

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Q: Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is…

A: Given that x^4=x for any x in R. To prove R is commutative.

Q: Enter the multiplicative inverse of the element [22]44 x + [17]44 in the ring ℤ44[x]. When you…

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

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Q: Enter the multiplicative inverse of the element [6]18 X+ [5]18 in the ring Z18[x]. When you enter…

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Q: Please use method from Abstract Algebra, Show that the polynomial 37a? + 30a + a4 = 1500 does not…

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Q: 1. State the Cancellation Axiom for the multiplication. 2. Show, justifying each step, that the…

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Q: The triple ⟨911Z, +, ⋅⟩ forms a commutative ring. It is also an integral domain. However, it is not…

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Q: Please use method from Abstract Algebra, Show that the polynomial 37a + 30a + a4 = 1500 does not…

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

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: a. Find the elements of Zg that have multiplicative inverses.

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Q: n the ring Z5 × Z7, determine −(3, 6) and (3, 6)−1 .

A: Here given ring is R = Z5 × Z7  We know that properties of ring R  If (a,b) is element of R then, i)…

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

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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,…