I am reading R.Y Sharp's book: "Steps in Commutative Algebra".

On page 6 in 1.11 Lemma, we have the following:[see attachment]

"Let S be a subring of the ring R, and let be a subset of R.

Then is defined as the intersection of all subrings of R which contain S and .

Thus, is a subring of R which contains both S and , and it is the smallest such subring of R in the sense that it is contained in every other subring of R that contains S and .

In the special case in which is a finite set we write as .

In the special case in which S is commutative, and is such that for all we have

.............................................. (1)

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Then on page 7 Sharp writes:

Note that when R is a commutative ring and X is an indeterminate, then it follows from 1.11 Lemma that our earlier use of R[X] to denote the polynomial ring is consistent with this new use of R[X] to denote 'ring adjunction'.

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Now in the polynomial ring R[X] we take a subset of ring elements and use an indeterminate x (whatever that is?) to form sums like the following:

.................................................. ...... (2)

My problems are as follows:

(a) It looks like (1) and (2) have the same structure BUT is a member of the ring R, and also the subring S whereas x is not a member of R but is an "indeterminate" [maybe I am overthinking this and it does not matter??] Can someone please clarify this matter?

(b) Again, (1) and (2) seem to have the same structure BUT is just a subset of R - whereas are elements of a subring. Does this matter? Can someone please clarify?

(c) Sharp specifies that S has to be commutative - but why? I cannot see how this is needed in his Proof on the bottom of page 6. Can someone help.

I would be grateful if someone can clarify the above.

Peter