6.1. A RECOLLECTION OF RINGS
267
We gave a number of examples of subrings in Example
1.11.5
. You
are asked to verify these examples, and others, in the Exercises.
For any ring
R
and any subset
S
±
R
there is a smallest subring of
R
that contains
S
, which is called the
subring generated by
S
. We say that
R
is generated by
S
as a ring if no proper subring of
R
contains
S
.
A “constructive” view of the subring generated by
S
is that it consists
of all possible ﬁnite sums of ﬁnite products
˙
T
1
T
2
²²²
T
n
, where
T
i
2
S
. In particular, the subring generated by a single element
T
2
R
is
the set of all sums
P
n
i
D
1
n
i
T
i
. (Note there is no term for
i
D
0
.) The
subring generated by
T
and the multiplicative identity
1
(assuming that
R
has a multiplicative identity) is the set of all sums
n
0
1
C
P
n
i
D
1
n
i
T
i
D
P
n
i
D
0
n
i
T
i
, where we use the convention
T
0
D
1
.
The subring generated by
S
is equal to the intersection of the family
of all subrings of
R
that contain
S
; this family is nonempty since
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 Fall '08
 EVERAGE
 Algebra

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