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: The characteristics of local rings is either 0 or prime.

A: Let R be a local ring (with 1). Suppose char R = n ≠0. Let, if possible, n be divisible by two…

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

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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: Define a new addition and multiplication on Q by a b = a + b+ 1 and a b = ab + a +b, where the…

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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: An example on a finite non- commutative ring without identity is:

A: Matrices are noncommutative  So we can choose set of matrices  Not if we take Mn[R] then this is non…

Q: Define commutative ring.

A: Answer:

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

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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: Exercise 36. Let A,B, and C be ideals of a commutative ring R. Prove the following: (1) A.(B.C)…

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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: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

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

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Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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Q: Q 10 Which one of the following is not a ring? (a) (Z, +, ·) (b) (M4x4(R), +, ) (c) (Zs,O5, 85) (d)…

A: We know that of G is group then G is follow closure property .  G is follow associative property.…

Q: Construct the multiplication table for the ring Z/5Z.

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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: (c) Let R be a finite commutative ring with identity. Prove that an ideal in R is prime if and only…

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Q: Define a new addition ⊕ and multiplication ⊙ on ℚ by               a ⊕ b = a + b + 1 and a ⊙ b = ab…

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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: 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: Show that every ring can be embedded in a ring with identity.

A: We have to show that every ring can be embedded in a ring with identity.

Q: M„(Z) D This Option

A: Clearly the ring Mn(F) is infinite iff F is an infinite field.