cg. 2. Let a, b, c, f, and g be elements of a commutative ring R with a‡OR. Prove that if a divide both b and c, then a divides bf + (Note that in a commutative ring R where s and t are elements in R, then s divides t if there is an element b R such that t=s*b AND S0R.) Proof/

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

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A: Since you have asked multiple questions, we will solve the first Question for you. If youwant any…

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Q: 38

A: To check: Whether Z6 is a subring of Z12 or not.

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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…

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Q: The ring Z-[i] has no proper ideals True False

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Q: by example thai it is an sible to find ideals A,B,C R Such that a

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Q: Q1: what's the relation between prime and semiprime ideals.

A: As per the guidelines we are supposed yo solve the first question.kindly post another questions…

Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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Q: Q28: Define the concept of field. Is (R-{0},+,.) field?

A: Dear Bartleby student, according to our guidelines we can answer only three subparts, or first…

Q: X a b с a b a C

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Q: et n be a positive integer and a, b₁,b₂, ..., bn be elements of a commutative ing with unity R.…

A: Let n be a positive integer and a,b1,b2,…,bn be elements of a commutate ring with unity R. We will…

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Q: Prove that the triple Z. , 0 has no zero divisors for prime p E P.

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

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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:…

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

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

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

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Q: Q1: Find all tHhe idempotent and nilpotent e le ment of the ring Z5

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Q: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).

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