632 MATHEMATICS: S. BOURNE PROC. N. A. S. It follows that O(u, x) satisfies the equation 0(u, X) = 1 + F1 ee(l/u+l/v)(x-8) X(y) (u, y) 0(v, y) dyl . (3.9) This leads to a feasible computational solution which we shall discuss in further detail subsequently. 1 V. A. Ambarzumian, "On the Scattering of Light by a Diffuse Medium," Compt. rend. Doklady Acad. sci. U.R.S.S., 38, 257, 1943. 2 S. Chandrasekhar, Radiative Transfer (London: Oxford University Press, 1950). 3 R. Bellman and T. E. Harris, "On Age-dependent Binary Branching Processes," Ann. Math., 55, 280-295, 1952; "On Age-dependent Branching Processes," these PROCEEDINGS, 34, 601-604, 1948. 4 S. Janossy, Cosmic Rays (London: Oxford University Press, 1950). 6 J. Hadamard, "Le Principe de Huygens," Bull. Soc. Math. France, 52, 610-640, 1924. 6 E. Hille, "Functional Analysis and Semi-groups," Am. Math. Soc., 1948. 7 R. Bellman, "The Theory of Dynamic Programming," Bull. Am. Math. Soc., 60, 503-516, 1954. 8 G. G. Stokes, Mathematical and Physical Papers, Cambridge, 1904, Vol. IV: On the Intensity of the Light Reflected from or Transmitted through a Pile of Plates, pp. 145-156. Stokes' method was rediscovered independently by R. K. Luneberg (The Propagation of Electromagnetic Plane Waves in Plane Parallel Layers [Research Report No. 172-3, June, 1947, New York University, Washington Square College]), and R. M. Redheffer ("Novel Uses of Functional Equations," J. Rat. Mech. Anal., 3, No. 2, 271-279, 1954). ON MULTIPLIcATIVE IDEMPOTENTS OF A POTENT SEMIRING* BY SAMUEL BOURNE UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA Communicated by H. S. Vandiver, June 26, 1956 1. INTRODUCTION A semiring is a system consisting of a set S and two binary operations in S called addition and multiplication such that (a) S together with addition is a semigroup; (b) S together with multiplication is a semigroup; (c) the left- and right-hand distributive laws a(b + c) = ab + ac and (b + c)a = ba + ca hold. Semigroup is used in the sense of a closed associative system. R. Brauer men- tioned to Vandiver' that Dedekind concerned himself with such an algebraic sys- tem without giving the semiring a formal definition. The latter was first intro- duced by Vandiver.2 The simplest example of a semiring is the system of count- ing numbers relative to ordinary addition and multiplication. Vandiver also gave an example' of a semiring which cannot be imbedded in a ring. Another sig- nificant example is the semiring of finite and infinite cardinal numbers' which cannot be imbedded in a ring. Independently, this system came to us4 naturally as the totality of endomorphisms e of an arbitrary additive semigroup S." A nonimbeddable semiring arises naturally when we take the totality of endomor- Downloaded by guest on October 5, 2021 Vot, 42,4MAT19356 HEMA TICS: S. BOUlRlNE (633: phisms e of a nonimbeddable additive semigroup. If both semigroups of a semi- ring are commutative, we say that the semiring is commutative. An important example of a commutative semiring is a distributive lattice.6 In particular, the set of ideals of a commutative ring forms a commutative semiring relative to the operations sum and product.' Our main purpose in this paper is to obtain an analogue for semirings of the fundamental lemma of structure theory in rings, that, if the left ideal I of a ring T) with minimum condition is non-nilpotent, then f contains a multiplicative idem- potent.8 Clifford9 proved that in a semigroup e without nilpotent ideals (0) and in which every two-sided ideal contains at least one left and at least one right minimal ideal of 5, every left ideal L # (0) of e contains an idempotent element $ (0). We now show that this theorem remains true for semirings. This paper has benefited materially from discussion and correspondence with A. H. Clifford, of Tulane University, and H. Zassenhaus, of McGill University. 2. ADDITIVE AND MULTIPLICATIVE IDEMPOTENTS Definition 1: An additive identity, called zero, is an element 0, such that 0 + s = s + 0 = s for all s in S. Definition 2: An additive idempotent of a semiring S is an element a such that a + a = a and a multiplicative idempotent is an element m such that m2 = m, m 5 0. A distributive lattice is an example'0 of a commutative semiring in which each element is both an additive and a multiplicative idempotent. If S possesses a zero, then aO = a(0 + 0) = 2aO and Oa = 20a are additive idem- potents which are not necessarily equal. Example 1: Let 52 be the noncommutative additive semigroup of order 2 with elements 0, 1 whose additive table is + 0 1 0 0 1 1 0 1 Let e be the totality of endomorphisms of 52. Then e = O, I, A, B}, where (0, 1)0 = (0, 0), (0, 1)I = (0, 1), (0, 1)A = (1, 1), and (0, 1)B = (1, 0). The addition and multiplication tables in the semiring A, are, respectively, + O I A B la O I A B 0 O I AB 0 OO AA I O I A B I O I A B A O I A B A O A A O B O I A B B O B A I Since OA = A 5 0 = AO, we have two distinct additive idempotents. Example 2: Let S2 be the semiring of order 2 with elements 0, 1 whose addition and multiplication tables are, respectively, + 0 1 0 1 0 0 1 0 1 1 1 1 I1 1 11 Downloaded by guest on October 5, 2021 634 MATHEMATICS: S. BOURNE PROC. N. A. S. In this example S2 possesses a zero element 0 in which 02 $ 0. We shall assume that S possesses a zero element 0 and Os = sO = 0, for all s in S. Definition 3: A division semiring is a semiring in which the elements $0 form a multiplicative group. 3. TWO-SIDED IDEALS CONTAINING MINIMAL RIGHT IDEALS For the sake of completeness, we repeat the definition of right ideal given else- where.4 Definition 4: A right ideal S is a subset R of S containing zero such that if ri and r2 are in R, then r1 + r2 is in R, and if r is in R and s is any element of S, then rs is in R. Definition 5: The sum over a given set A of right ideals R1 is the smallest right ideal containing all the given right ideals. It consists of all finite sums En, where each ri belongs to one of the right ideals Ri. We denote this right ideal by ERj. This addition is both commutative and associative in the widest sense. In the case addition, in the semiring S, is commutative, our definition of the sum of right ideals becomes identical with the standard definition in ring theory; hence R1 + R2 is the set of sums ri + r2 with ri in R1 and r2 in R2. Definition 6: The product R1R2 of the right ideals R1 and R2 of a semiring S is the set of all finite sums Eriir2i with rjj in R1 and r2i in R2. This multiplication of ideals is associative but not necessarily commutative. Also, the general distributive law ZREZQj = ZRiQj holds. i,j Let us mention the fact that the intersection of any number of ideals of the same kind is again an ideal of the same kind. Definition 7: A right ideal N is said to be nilpotent if a power NP is equal to the zero ideal (0). Definition 8: A semiring S is said to be potent if it contains no nonzero nilpotent right ideals and left ideals. LEMMA 1. If R is a minimal right ideal of a semiring S, then tR, where t is in S, is either a minimal right ideal S or tR = (0). Proof: tR is a right ideal. Let A be a right ideal contained in tR $ (0) and R1 the set of all elements ri of R such that tr1 is in A. Since t(rlk + rie) = trlk + trie and t(rlk)S = t(rlk)S are in A, we have that ruk + rne and rlkS are in R1. This implies that R1 $ (0) is a right ideal contained in R. Therefore, R1 = R, tRi = tR C A, and A = tR. This proves that tR is minimal. THEOREM 1. IfI is a two-sided ideal containing a minimal right ideal of S, the sum J of all minimal right ideals of S contained in I is a two-sided ideal of S. If I is minimal, then it is a sum of minimal right ideals of S. Proof: Since J is the sum of right ideals Ri, it is a right ideal. We show that it is also a left ideal. Let j e J and s e S; then sj = sEri = Esri, where rj belongs to some minimal right ideal Ri contained in I. Now sri e sRi, and sR1 is, by Lemma 1, a minimal right ideal contained in I, or sRj = (0). Hence, in either case, sj e J and J is a left ideal. Obviously, if I is minimal, I = J. THEOREM 2. If M is a minimal two-sided ideal of a potent semiring S, containing a minimal right ideal of S, then every minimal right ideal of S contained in M is a minimal right ideal of Al. Downloaded by guest on October 5, 2021 VOL. 42, 1956 MATHEMATICS: S. BOURNE 635 Proof: Let R $ (0) be a minimal right ideal of S contained in M, and A a right ideal $ (0) of M contained in R.