Q1: Suppose R is a ring with unity 1, a E R and a? = 1, let S = {ara : r E R}. Prove that S is a subring of R.

Q: 5. Let R be a ring (not necessarily commutative). Prove that 0 -r = 0 and -x = (-1) · x for every x…

Q: Let R be a ring and A be an idea of R where r, sER Explain why rs+ A=sr+A is equivalent to rs – sr E…

Q: 5. Let R be a ring and r1,., r, € R. Prove that the subset (r1,., "n) = {Airı + Anrn | A1,. An E R}…

Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

Q: Let R be a ring, and let ICR be an ideal. Prove that R/I is commutative if and only if xy - yx € I…

Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

Q: 1. Let a be an element of a ring R. Let S = {x €= R | ax = 0}. Show that S is a subring of R. Show…

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

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

Q: 1. Prove that if S is a subring of a ring R then the following are true: (a) Os = OR (b) if 1R € S,…