MHB Ring Theory: Proving Subrings and Ring Generation

[Poirot1]

Two questions

(1)For R a ring and A a subset of R, let s(A) denote the set of all subrings of R that contain A (including R itself). Show that the intersection of all these subrings is itself a subring of R.

(2)Suppose that 1 is not equal to 0 in R. Show that the sets ∅, {0} and {1} all generate the same ring in R.​

Thanks

 

[caffeinemachine]

Two questions

(1)For R a ring and A a subset of R, let s(A) denote the set of all subrings of R that contain A (including R itself). Show that the intersection of all these subrings is itself a subring of R.

(2)Suppose that 1 is not equal to 0 in R. Show that the sets ∅, {0} and {1} all generate the same ring in R.​

Thanks

1) Check whether the following conditions are met:
a) 0 is in s(A)
b) a - b is in s(A) whenever a and b are in s(A)
c) ab is in s(A) whenever a and b is are in s(A)