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 $\displaystyle \Gamma $ be a subset of R.

Then $\displaystyle S[ \Gamma ] $ is defined as the intersection of all subrings of R which contain S and $\displaystyle \Gamma $.

Thus, $\displaystyle S[ \Gamma ] $ is a subring of R which contains both S and $\displaystyle \Gamma $, 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 $\displaystyle \Gamma $.

In the special case in which $\displaystyle \Gamma $ is a finite set $\displaystyle \{ \alpha_1, \alpha_2, ... ... , \alpha_n \} $ we write $\displaystyle S[ \Gamma ] $ as $\displaystyle S [ \alpha_1, \alpha_2, ... ... , \alpha_n ] $.

In the special case in which S is commutative, and $\displaystyle \alpha \in R $ is such that $\displaystyle \alpha s = s \alpha $ for all $\displaystyle s \in S $ we have

$\displaystyle S[ \alpha ] = \{ \ {\sum}_{i = 0}^{t} s_i \alpha^i : t \in {\mathbb{N}}_0 \ s_0, s_1, ... ... , s_t \in S \} $ .............................................. (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 $\displaystyle a_1, a_2, ... ... , a_n \in R $ and use an indeterminate x (whatever that is?) to form sums like the following:

$\displaystyle a_n x^n + a_{n-1} + ... ... + a_1x + a_0 $ .................................................. ...... (2)

My problems are as follows:

(a) It looks like (1) and (2) have the same structure BUT $\displaystyle \alpha $ 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 $\displaystyle a_1, a_2, ... ... , a_n \in R $ is just a subset of R - whereas $\displaystyle s_0, s_1, ... ... , s_t $ 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