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Unformatted text preview: 278 6. RINGS Deﬁnition 6.2.24. The smallest left ideal containing a subset S is called
the left ideal generated by S . The smallest left ideal containing a single
element x 2 R is called the principal left ideal generated by x .
When R has an identity element the principal left ideal generated by x
is just Rx D frx W r 2 Rg: See Exercise 6.2.8
Proposition 6.2.23 and Deﬁnition 6.2.24 have evident analogues for
right ideals. The following is the analogue for two-sided ideals:
Proposition 6.2.25. Let R be a ring and S a subset of R. Let hS i denote
the additive subgroup of R generated by S .
RS R D fa1 s1 b1 C a2 s2 b2 C
(b) (c) C an sn bn W n 2 N ; an ; bn 2 Rg: Then RS R is a two-sided ideal.
hS i C RS C S R C RS R is the smallest ideal of R containing
S , and is equal to the interesection of all ideals of R containing
If R has an identity element , then hS i C RS R D RS R. Proof. Essentially the same as the proof of Proposition 6.2.23. I Deﬁnition 6.2.26. The smallest ideal containing a subset S is called the
ideal generated by S , and is denoted by .S /. The smallest ideal containing
a single element x 2 R is called the principal ideal generated by x and is
denoted by .x/.
When R has an identity element, the principal ideal generated by x 2
.x/ D fa1 xb1 C a2 xb2 C C an xbn W n 2 N ; ai ; bi 2 Rg: See Exercise 6.2.9. When R is commutative with identity, ideals and left
ideals coincide, so
.x/ D Rx D frx W r 2 Rg:
The ideal generated by S is, in general, larger than the subring generated by S ; for example, the subring generated by the identity element
consists of integer multiples of the identity, but the ideal generated by the
identity element is all of R. ...
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- Fall '08