By Neil Hindman; Dona Strauss

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A) L is a left ideal of S if and only if ; ¤ L Â S and SL Â L. (b) R is a right ideal of S if and only if ; ¤ R Â S and RS Â R. (c) I is an ideal of S if and only if I is both a left ideal and a right ideal of S. An ideal I of S satisfying I ¤ S is called a proper ideal of S. Sometimes for emphasis an ideal is called a “two sided ideal”. We often deal with semigroups in which the operation is denoted by C. 4 Ideals seem awkward for someone who is accustomed to working with rings. That is, a left ideal L satisfies S C L Â L and a right ideal R satisfies R C S Â R.

Let f D rt s. S/. Also, f e D rt se D rt s D f and ef D ert s D rt s D f so f Ä e so f D e. Thus Se D Sf D S rt s Â S s Â L. (c) Let L be a left ideal with L Â Se. We show that e 2 L (so that Se Â L and hence Se D L). Pick an idempotent t 2 L, and let f D et . Then f 2 L. Since t 2 Se, t D t e. Thus f D et D et e. S/. Also ef D eete D ete D f and f e D et ee D et e D f so f Ä e so f D e and hence e 2 L. (d) This follows from (a), (b), and (c). We now obtain several characterizations of a group.

B) implies (a). Let G D ¹x 2 S W xe D x and there is some y 2 S such that ye D y and xy D yx D eº. It suffices to show that G is a group with identity e. To establish closure, let x; z 2 G. Then xze D xz. Pick y and w in S such that ye D y, we D w, xy D yx D e, and zw D wz D e. Then wye D wy and xzwy D xey D xy D e D wz D wez D wyxz. Trivially, e is a right identity for G so it suffices to show that each element of G has a right e-inverse in G. Let x 2 G and pick y 2 S such that ye D y and yx D xy D e.