Zeleznikow, J.

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeJohn ZeleznikowOn regular ring-semigroups and semiringsCommentationes Mathematicae Universitatis Carolinae, Vol. 25 (1984), No. 1, 129--139Persistent URL: http://dml.cz/dmlcz/106284Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1984Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITAT.S CAROLINAE 25,1 (1984) ON REGULAR RING-SEMIGROUPS AND SEMIRINGS J. ZELEZNIKOW Abstract? Regular and orthodox ring-semigroups and semi-rings arT^EaracteriBed, as well as ring-semigroups with chain conditions on idempotents and principal idea l s . Congruences on addit ively regular semirings are also considered. Key words: Ring-semigroup, addit ive ly inverse semiring, orthodox semigroup, congruence, Green's re la t ions . Class i f icat ion! Primary 16A78 Secondary 20M75, 16A30 1» Introduction! In a semigroup (S,») we put B • { e e S i f2m%\ and V(x) - {aeS!X»a*x « x and a«x a • a* for a l l z « S . I f V(x) **> U , then the element x i s said to be regular. I f each element of S i s regular, then the semigroup S i s said to be regular. I f S i s a regular semigroup, and E i s a subse-migroup of S, then S w i l l be said to be an orthodox semigroup. A regular semigroup in which e»f » f*e for a l l e , f e E , i s said to be an inverse semigroup. We use the def ini t ions and notation of C13. A semigroup (S,») i s a ring-semigroup i f there e x i s t s a binary operation +:Sx.S—> S such that (S , + ,«) i s a ring. In 1121, the structure of orthodox ring-semigroups was considered. Such semigroups are inverse. In the proof of th i s theorem, the concept of an addit ivelv inverse semiring i s re-quired. 129 -Definition 1i A triple (St+tO ia a semiring if S is a set, and +t• are binary operations satisfying (i) (St+) is a semigroup, (ii) (S,-) is a semigroupt (iii) a*(b + c) « a.b + a*o, (a + b)*o • a#c + b*c 9 for all atbtoc S. Definition 2t A semiring (S t+ tO is said to be an additi-vely inverse semiring if (St+) ia an inverse semigroup. The following theorem of Karvellas allows us to prove many results for additlvely inverse semirings. Result 3: (173 Theorem 7.) In an additively inverse semiring (St+t-)t if as aSnSa for all ae S, then S is additively commutative (and hence a semilat-tice of groups). In a semiring (S,+t») we put E » -Cxe S:x + x « xi and C-l <. i E • (ec S:e*e * er. 2. Regular ring--semigroups: We can now prove: Result 4: (112.1 Theorem 9.) Let (St+tO be any additively inverse semiring in which (S,*) is regular. Then the following conditions are equivalent: ( i ) V e t f e EC O t (e*f « 0=* f** « 0 ) . ( i i ) Ve*E c ' 3 t V x e S , (e-x » 0 « > x-e - 0 ) . ( i i i ) . V n € «4 t V x e S , ( xn « 0~-> x « 0 ) . ( iv) V x 6 S , (x2 « 0 ^ x = 0 ) . ( T ) V x t y € S t (x*y « 0 -=£• y x = 0 ) . Further, each i s implied by (v i ) (S t«) ia orthodox. 130 -Example 5: In an arbitrary regular semigroup (S fO f oondi-tion (i) of Theorem 4 does not imply condition (ii)f and (S fO being orthodox does not imply condition (ii). To see this we may take any Brandt semigroup S » fliP(GfIfIf A ) in which t U > 2 . Thus this semigroup cannot be the multiplicative semigroup of an additively inverse semiring. Result 6: ( £121 Theorem 13-) In a regular ring-semigroup (S fO the following conditions are equivalent; ( i ) ( S , 0 i s orthodox* ( i i ) Ve , f e E , (e*f » 0 * - * £ • • - 0 ) . ( i i i ) V e eE, V x c S , ( e .x * 0 =-» x*e * 0 ) . ( iv) V r e l N , V x £ S f ( xr = 0 ~ - > x = 0 ) . (v) Vx&S, (x2 = 0 = » x = 0 ) . (vi) V x j c S , (x • y = 0 =-* y x = 0 ) . (v i i ) (3,*) i s inverse . Example 7: (i) Take (R, + ,0 to be a regular ring in which (R,0 is not orthodox. Set S = Rulal where a <j* R and define r + a = a + r = r, a + a - a = r*a = a»r for all rgR. Then (S,+,*) is a semiring in which ( S , » ) is regular and a is the ad-ditive and multiplicative £p-o of S. Hence (Sf + f O satisfies con-dition (v) of Result 6, but is not orthodox. (ii) Let (Sf+) be a semilattice with |Sl£2 and define A*j = x for all x fyeS. Then (S, + f«) is an additively inverse se-miring in which the multiplicative semigroup is orthodox but not inverse. Laliement (181 Theorem 4.6) has proved that a primitive re-gular ring-semigroup is a group with zero adjoined. In particu-lar, a corapletely-O-simple ring-semigroup is a group with zero adjoined. - 131 -Define a partial order on the set of idempotents E of a semigroup S by; f£e if and only if f - e-f-e. A nonzero idea-potent e is primitive in S if for f €E, 0-M £e implies f « e. The semigroup S satisfies Min - E if the minimum condition holds for E under the specified order; Max - E is defined dually. If x e S let J(x) « - I X J U X S U S X U S X S denote the principal (two-sided) ideal generated by x, and K x ) - 4yeJ(x);J(y) £ j(x)| the set of nongenerators of J(x). Then S is called completely semi simple if for each nonzero xgS, the Rees quotient semi-group J(x)/I(x) contains a primitive idempotent, in which oase every nonzero idempotent of J(x)/I(x) is primitive. We let Min - J signify the minimum condition on the set of principal ideals of S; Max - J is its dual. A ring is aemiprime if it contains no nonzero nilpotent (one-sided) ideals, and artinjan if it satisfies the minimum condition on right ideals, A ring is atomic if it is a (direct) sum of minimal right ideals* As a generalization of Lallement's theorem we have the fol-lowing result. Result 8; (151 Theorem 4.) For a semigroup S, the following conditions are equivalents (i) S is completely semisimple and satisfies Min - J. (ii) S is completely semisimple and satisfies Min - E# (iii) S is regular and satisfies Min - E. Furthermore, if S is a ring-semigroup, then (i),(ii) and (iii) are equivalent to each of the following conditions; 132 -(iv) (Sf+»*) is a semi prime atomic ring. (T) (Sf+,0 is a direct sua of dense rings of finite-rank linear transformations of vector spaces over division rings. Example 9i Whilst the equivalent conditions (i)t(ii),(iii) of Result 8 imply that S is regular with Min - J, the converse does not hold, even for rings with identity. To see this, consi-der the full ring of linear transformations of an infinite-di-mensional vector space. This ring is regular (t93» Theorem 7*3) with Min - J (POl, Theorem 1.4.2) hut does not satisfy Min - E, since the projections onto an infinite descending chain of sub-spaces give rise to an infinite descending chain of idempotents. Result 10I ([5] Theorem 5.) For a semigroup Sf each of the following conditions implies the next.. (i) S is completely semisimple and satisfies Max - J. (ii) S is completely semisimple and satisfies Max - E. (ill) S is regular and satisfies Max - E. Furthermore. if S is a ring-semigroup, then conditions (i)f (ii) and (iii) are equivalent to each other and to the condition; (iv) (Sf+f*) is a aemlprime artinian ring i.e. a finite direct sum of full matrix rings over divi sion rings. Example 11; (i) The bicyclic semigroup /3(p*a_) * » <pfq:pq « I4~qp> is regular and satisfies Max - E hut not Min - E ([13 Theorem 2.53)» Moreover it is not completely semi-simple. Thus in Theorem 10, condition (iii) does not imply condi-tion (ii) for non~ring-semigroups-(ii) Let C he the chain of length n, n*2. Suppose these chains have a common zero element 0. Take E to be the 0-direct union of C , n£ 2. Then E is a semi lattice satisfying Max - E and 133 Min - E. The Munn semigroupf Tg of E (£6.3 Section V.4) is an in-Terse semigroup with B as its semilattiee of idempotenta (and thus is completely aemisimple by Thtortm 8) hut does not satis-fy Max - J. Thus in Theorem 10, (ii) =-> (i) is not valid for non-ring-semigroups. (iii) [2] Examples (a)9(h) page 803 give examples of regu-lar ring-semigroups which: (a) have only two principal ideals but do not satisfy Max - E or Min - E, (b) are eompletely semisimple but do not satisfy Max - J, Min - J, Max - E or Min - E. 3. Congruences on regular semirings! Semirings in which the additive semigroup is inverse and the multiplicative semi-group Is regular (and henoe the additive semigroup is a semilat-tiee of abelian groups) are oonsldered in C113,C131. These pa-pers also consider the oaee in which the multiplicative semi-group is simple or 0-simplt. Result 12i (Ml) In a semiring (Sf+fO the additive Green s relations e£9Tl9X,Zb 9T are congruences on the multiplicative semigroup (S fO. • semigroup (Sf») is said to be congruence-free if the on-ly congruence relations on S are i* and SxS. Thus a congru-ence-fret semigroup is simple or 0-simplt, since if I is an 1-deal of Sf J>j defined by J-»x • (Ixl) o lSi is a congruence relation on S. A band (S,*) is left (right) regular if axa « ax (axa -« xa) for all a,xeS. 134 -Lemma 13i Take (St») to be a regular semigroup on whioh £ ( & ) is trivial. Then S is a right (left) regular band i.e. a semilattice of left [right! sero semigroups. Proofi Take x taeS and a*£V(a). Then a^ta'a and thus a • a "a. Hence a * a(a'a) » a. How S axafiS xa - S xaxa e 9 S axa and so S axa • S za i.e. axa«£xa * Thus axa » xa for all a txcS. Q Corollary 14t Take (St») to be a regular semigroup on whioh 2) is trivial. Then (St*) is a semilattice. Proofi Since «C» & » 4 g t S is both a left and right re-gular band and henoe a semilattice. Q A semiring (St+t») is said to be completely simple if the additive semigroup is completely simple and the multiplicative semigroup le either completely simple or completely 0-simple. Theorem 15! ([13) Theorem 24)• Take (St+t») to be a comple-tely simple semiring. (i) If the multiplicative semigroup is completely 0-simp-le, then the semiring is a division ring. (ii) If the multiplicative semigroup is completely simple. then the additive semigroup is a rectangular band and the mul-tiplicative semigroup is a product of two completely simple se-migroups S » I x A and the operations on the semiring S are given by (i,a) + (d,<u,) - (i,(U,) (i tAMj t<u,) » (i«jtA.<u.) for all i..1 €l. A f f-tcA. Theorem 16! Take (St+t») to be a semiring in whioh the ad-ditive semigroup is regular and the multiplicative semigroup is - 135 -congruence-free. Then the additive semigroup is either a group. a aemilattice. a left zero band or a right zero band. Proofs By [3] Lema 2 (i), the set B W is an ideal of (S,*). Since (S,») is simple or O-simple, E^+3 » 10 J or S W « « S. Since (3,+) is a regular semigroup, it is either a group or a band* Because T is a congruence on the multiplicative semigroup (S,0» T - lg or Tm SxS. (i) In the case .T« 4 g, then (S,+) is a semilattlce by Corollary 14, since 2) £ (H. (ii) When (T-* SxS, (S,+) is a simple semigroup. (a) 3C- tf« Jt»2> -* 1s. Then (S,+) is a simple semilattice and thus the trivial group. (b) % »S£ » l g f fl.S - SxS. Then (S,+) is right simple and a band. Thus, by LU Theorem 1.27, S is the direct product of a group and a right zero band and thus is a right zero band since *K -» Ig. (C) %m%m 1 g t SC«(B « SxS. By symmetry, (S,+) is a left zero semigroup. (d) X M SxS. In this case (S,+) is a group. D Example 17: We provide examples of semirings in which the additive semigroup is regular and the multiplicative semigroup Is congruence-free, as in Theorem 16. (i) Take (S,+,») to be the two element field. Then (S,0 Is congruence-free. Here (S,+) is a group. (ii) The two-element chain has as its multiplicative se-migroup a congruence-free semigroup. Here (S,+) is a semilatti-ce. - 136 -( i i i ) Take (S,») to be any congruence-free semigroup. Define the binary operation + s S x S - » S by x + y «• x for a l l x t y € S . Then (S t + t O i s a semiring in which the additiTe se -migroup i s a le f t -zero band. Theorem 18: Take (S t+ t») to be a semiring in which (S t+) i s a regular semigroup and ( S , 0 has a unique non-tr iv ia l con-gruence. Then the additive semigroup i e either a group, a »*-milatt ice of groups, a semilatt ice of l e f t zero bands, a semi-l a t t i c e of ri^ht zero bmnds. a l e f t group or a right group. Proof? Denote by p the non-tr iv ia l congruence on (S t *) . Since 1C s £ £ © & 1 and X<*%G®&'ftm% have that e i t -her l f s . f t o r 3 f c s S 6 . W e shal l only consider the cases in which £ £ & , since the resul t s for % SkX w i l l follow by symmetry. (D se - . » - a - 1 S . By Corollary H t (St+) is a semilattice. (ii) %»j£.&«ffl-.<r. Clearly, a regular semigroup in which «6 » % is a semilatti-ce of groups. (iii) *X • & • % •% -pC-T.SxS. In this case, (St+) is a semilattice of groups and also sim-ple since Cf» SxS, Hence (St+) Is a group. We now consider the case in which o£ c 4t * (iv) 13 •%• £ £ fi, • ® Q<T & S*S. Since X is trivial, by Lemma 13, S la a right regular band, i.e. a semilattice of right zero semigroups. The other cases were considered in Theorem 16 or follow by symmetry. 137 R e f e r e n c e s L13 A.H. CLIFFORD and G.B. PRESTOH (1961), The ftlgeЪralo tћe-ory of semigroups, Vol. 1 (Ameгioan Mathematical Society. Mathematical Surveys 7 ) . [23 K.R. GOODEARL and D. HANDELBIAN (1975), Simple se l f- inЗec-t ive r ings, Comm. AlgeЪra 3 ( 9 ) , 797-834. LЗl M.O.P. GRILLET (1970), SuЪdivision rings of a semiring, Fund. Math, 67, 67-74. £43 M.O.P. GRILLET (1975), Semirings with a oompletely вimplo additive semlgroup, J. Austral. Math. Soc. 20, 257-267. L53 J. HANNAH, J . S . RICHARDSON and J. ZELБZNIKOW (1980), Com-plete ly semisiraple ring-semlgroups, J. Austral. Math. Soc. 30, 150-156. [63 J.M. HOWIE (1976), An introduction to semigroup theory, Academic Press, Ino. , London. [73 P.H. KARVELLAS (1974), Inversive serairings, J. Austral. Math. Soc. 18, 277-287. L83 G. LALLEMENT (1967), Demi-groups regul iers, Annali mat. pura. appl. 77, 47-130. L93 H.H. McCOY (1964 ), The theory of r ings iмaomillan, Hew York). LЮЗ M. PETRICH (1974 ), Rings and seraigroups (Leoture Notes i a Mathematics 380, Spгingeг-Veгlag, Berlin,Heidel-Ъerg, New York). LHЗ H.J. WEINERT (1983), On 0-simple semirings, seraigroup se-mirings and two kinds of d iv is ion semirings (to appear)• L123 J. ZELБZNIKOW (1980 ), Orthodox semirings and ringв, J. Austral. Math. Soc. 30, 50-54. L13З J. ZELEZNIKOW (1981), Regular semirings, Semigroup forum 23, 119-136. 138 -Department of Mathematica Michigan State Univerвity East Lanaing, Michigan 48824 United Stateя of Aшerioa (Oblátům 17.11. 1983) - 139 -

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On regular ring-semigroups and semirings

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On regular ring-semigroups and semirings

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Institute of Mathematics AS CR, v. v. i.

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