What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S: Prove or disprove: If a ∈ S is a unit, then a is also a unit in R

Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

A:

Q: The intersection of the collection of al the subrings containing a given subset M of a ring R is the…

A: Let R be a ring of which M is a given subset.  Let, C = { R1 , R2 ,. . . . . . . . Rk } be the…

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: A shopper buys cat food in bags of 3 lbs. Her cat eats 3434 lb each week. How many weeks does one…

A: Given that A shopper buys cat food in bags of 3 lbs. Her cat eats 3/4 lb each week

Q: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…

A:

Q: et R be a ring and I be an ideal of R. Let p: rs-sr is in A where r,s is an element of R q:…

A:

Q: Let R be a ring with unit. Define 2R ≔ 1R + 1R. In most rings, 2R ≠ 1R. Figure out what the…

A: Please check the answer in next step

Q: Let R be a ring with the left identity e. Suppose, for each nonzero element a in R, there exists b…

A: R is a ring with the left identity e.For each nonzero element a in R, there exists b in R such that…

Q: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

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

A:

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: 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…

A:

Q: 1.20. Let R be a ring with identity, and let J be an ideal of R. Assume that J contains a unit of R.…

A:

Q: (a) Give an example showing the following characterisation of a subring is incorrect, and explain…

A:

Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

A:

Q: Let R be a ring. If r3=r for all r in R, prove that R is communicative.

A:

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…