Ring Theory: Proving Subrings and Ring Generation

[Poirot1]

Two questions

(1)For R a ring and A a subset of R, let s(A[FONT=CMR10]) denote the set of all subrings of [FONT=CMMI10][FONT=CMMI10]R [FONT=CMR10][FONT=CMR10]that contain [FONT=CMMI10][FONT=CMMI10]A [FONT=CMR10][FONT=CMR10](including [FONT=CMMI10][FONT=CMMI10]R [FONT=CMR10][FONT=CMR10]itself). Show that the intersection of all these subrings is itself a subring of R.

(2)[FONT=CMSY10][FONT=CMSY10]Suppose that 1 is not equal to 0 in R.[FONT=CMR10][FONT=CMR10] Show that the sets [FONT=CMSY10][FONT=CMSY10]∅[FONT=CMR10][FONT=CMR10], [FONT=CMSY10][FONT=CMSY10]{[FONT=CMR10][FONT=CMR10]0[FONT=CMSY10][FONT=CMSY10]} [FONT=CMR10][FONT=CMR10]and [FONT=CMSY10][FONT=CMSY10]{[FONT=CMR10][FONT=CMR10]1[FONT=CMSY10][FONT=CMSY10]} [FONT=CMR10][FONT=CMR10]all generate the same ring in [FONT=CMMI10][FONT=CMMI10]R[FONT=CMR10][FONT=CMR10].​

Thanks

 

[caffeinemachine]

Two questions

(1)For R a ring and A a subset of R, let s(A[FONT=CMR10][FONT=CMR10]) denote the set of all subrings of [FONT=CMMI10][FONT=CMMI10]R [FONT=CMR10][FONT=CMR10]that contain [FONT=CMMI10][FONT=CMMI10]A [FONT=CMR10][FONT=CMR10](including [FONT=CMMI10][FONT=CMMI10]R [FONT=CMR10][FONT=CMR10]itself). Show that the intersection of all these subrings is itself a subring of R.

(2)[FONT=CMSY10][FONT=CMSY10]Suppose that 1 is not equal to 0 in R.[FONT=CMR10][FONT=CMR10] Show that the sets [FONT=CMSY10][FONT=CMSY10]∅[FONT=CMR10][FONT=CMR10], [FONT=CMSY10][FONT=CMSY10]{[FONT=CMR10][FONT=CMR10]0[FONT=CMSY10][FONT=CMSY10]} [FONT=CMR10][FONT=CMR10]and [FONT=CMSY10][FONT=CMSY10]{[FONT=CMR10][FONT=CMR10]1[FONT=CMSY10][FONT=CMSY10]} [FONT=CMR10][FONT=CMR10]all generate the same ring in [FONT=CMMI10][FONT=CMMI10]R[FONT=CMR10][FONT=CMR10].​

[FONT=CMR10][FONT=CMR10][FONT=CMR10]

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)