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462 ERRATA PROC. N. A. S.

ERRA TA Dr. R. E. Johnson has called to my attention that Theorem 1 of my paper "The of a " in these PROCEEDINGS, 37, 163-170 (1951) is in part incorrect. This theorem should read as follows: If I is an of a semiring S, then S is homomorphic to the difference semiring S = S - I. SAMUEL BOURNE Downloaded by guest on September 26, 2021 Vol.. 37, 1951 31MA THEMA TICS: S. BOURNE 163

a (W1(%qw(2) ..W(n)) However, the coefficient of u? is 1. Hence all the coefficients are rational . Letting u2 = 1, ua = U4 = ... = = Owe seev(2) for (i = 1, 2, ...,n) are all algebraic numbers. Similarly, we have vj/) for i, j = 1, 2, ..., n are all algebraic numbers. Let K = R(vP2), 3(l)M, ..., P.(l)), where R is the rational numbers (i.e., K is the least algebraic extension of R containing v2M'), . ..,vI (1)). As ul + P2(1)u2 + . . . + Pn(l)un $ Ounless u, = u2 = ... = un =O, we see that (K:R) = n' n. Hence, consider the n' different of K leaving R fixed. As they leave R fixed they leave the coefficients of the u's in the expansion of (9) fixed. By Hilbert's theorem on the uniqueness of the irreducible factors of a in n variables, we see that the n' iso- of K must merely permute the linear factors on the right side of (9). As all isomorphisms of K act differently on ul + u2 2(1) + . . . + Unynv(), we see that ni' = n, and the linear factors on the right-hand side of (9) are the n - 1 different conjugates. This proves Theorem 1. IV. It is of some interest to note that to prove Theorem 1 we do not need the full hypothesis of discreteness. Actually we only need to note the fact that there exists one a p 0 which satisfies (3) and (4) for suffi- ciently large values of X. This modification has application to the critical of n dimensional paraboloids. However, in this present note, we cannot take up this problem in any detail. * Under the auspices of the Office of Naval Research. Bochner, S., Ann. Math., soon to be published. 'Siegel, C. L., Gottingen Nachr., 31, 25-31 (1922). 'Siegel, C. L., Math. Annalen, 85, 123-128 (1911). (See also footnotes 1 and 2 above.)

THE JACOBSON RADICAL OF A SEMIRING BY SAMUEL BOURNE INSTITUTE OF ADVANcED STUDY, PRINCETON, NEw JERSEY* Communicated by H. S. Vandiver, December 18, 1950 1. Introduction.-A semiring is a system consisting of a S together with two binary operations, called and , which forms a relative to addition, a semigroup relative to multiplication, and the right and left distributive laws hold. This system was first in- troduced by Vandiver.' He also gave examples2 of which cannot 164 MA THEMA TICS: S. BO URNE PROC. N. A. S.

be imbedded in a ring. Semirings arise naturally when we consider the set of of a commutative additive semigroup.3 Our purpose is to generalize the concept of the Jacobson radical of a ring4 to arbitrary semirings. In section 2 we define the concept of an ideal in a semiring S and develop the corresponding theorem for semirings. In section 3 we extend the definition of the Jacobson radical to arbitrary semirings, and in section 4 we obtain some properties of the Jacobson radical of a semiring. We conclude with a consideration, in sec- tion 5, of the Jacobson radical of semiring Sn This paper has profited greatly from discussion with C. A. Rogers, a colleague of mine at the Institute. 2. The Homomorphism Theorem.-We shall assume that the additive semigroup of S is commutative and that S possesses a . The latter assumption is not vital in the sense that if S lacked a zero element, we can easily adjoin one to S. Definition 1: An ideal of S is a subset I of S containing zero such that if il and i2 are in I, then il + i2 is in I, and if i is in I, and s is any element of S, then is and si are in I. We shall say that s1 is equivalent to s2 the ideal I, if there exist elements ii and i2 of the ideal I such that si + i1 = S2 + i2. This definition is a translation to the additive notation of one given by Dubreil5 for a mul- tiplicative semigroup. This relationship is obviously an equivalence. The set of elements s equivalent to each other modulo the ideal I is called a of the ideal I. Relative to the usual definitions of an addition and multiplication the of an ideal I of a semiring form a semiring, called the difference semiring S - I. The correspondence s -' s defines a homo- of S into the difference semiring 3 = S -I. Conversely, if the semiring S is homomorphic to the semiring S', then the elements of S mapped into 0' of S' form an ideal I, and S' is isomorphic to 3 = S - 1, via the one-to-one correspondence s' < >. We have proved THEOREM 1. Ifthe semiring S is homomorphic to the semiring S', then S' is isomorphic to the difference semiring S - I, where I is the ideal of elements mapped into 0'. Conversely, if I is an ideal of 5, then Sis homomorphic to the difference semiring S - I. 3. The Jacobson Radical of a Semiring.-Definition 2: The element r of the semiring S is said to be right semiregular if there exist element r' and r in S such that r + r' + rr' = r + rr'. We notice that in the case S is a ring, this definition reduces to the one usually given for right quasi-regularity in a ring. THEOREM 2. A necessary and sufficient condition that the element r of S be right semiregular is that for any element s in S there exist elements s' and s' in S such that s + s' + rs' = s" + rs". Proof: If r is right semiregular, then r + r' + rr' = r" + rr". There- VOL. 37, 1951 MA THEMA TICS: S. BOURNE 165

fore, s + r's + rr's = s + r's + r(s + r's). On letting s' = r's and s' = s + r's, we obtain s + s' + rs' = s' + rs'. Conversely, we suppose that s + s' + rs' = s' + rs", for any s in S. In particular, r + s' + rs' = s' + rs' and r is right semiregular. Definition 3: The right ideal I is said to be right semiregular, if for every pair of elements ii, i2 in I there exist elements ji and j2 in I such that i4 + jl + iljl + jij2 = i2 + j2 + i1j2 + i2il- The elements ji and j2 are not unique, for this condition implies that +jl + iljl + i2j2 + j = i2 + j2 + i1j2 + i2jl + j, where j is any element inI, andii + (jl + j) + i1(j + j) + i2(j2 + j) = i2 + (j2 + j) + i(j2 +j) + i2(j1 + j). In particular, on substituting 42 = 0, we obtain that ii is right semiregular. LEMMA 1. If I and 1* are right semiregular ideals, then I + 1* is a right semiregular ideal. Proof: Since I is right semiregular, for every pair of elements il, i2 in I there exist elements j' and j2 in I such that i4 + Jl + iljl + i2j2 = i2 + j2 + i1j2 + i2Jl- If il* and i2* are in I*, then '** + il*jl + i2*j2 and i2* + i2*jl + ii*j2 are in I*. Since I* is right semiregular, there exist elements jl* and j2* in I* such that (i4* + il*jl + i2*j2) + jil + (il* + i *jl + i2*j2)jl* + (i2* + i2*jl + il*j2)j2= (i2* + 42*jl + il*j2) + j2* + (il* + il*jl + i2*j2)j2* + (i2* + i2*jl + il*j2)jl*. Therefore, (i4 + i) + (j& + ji* + jljl* + J2j2*) + (il + il*) (jl + jl* + jljl* + J2j2*) + (i2 + jg*)(j2 + j2* + jlj2* + j2jl*) = (il +Jl + iljl + i2j2) + [(il* + il*jl + i2*j2) + jl* + (il* + il*jl + i2*j2)jl* + (i2* + i2*jl + il*j2)j2*1 + (il +jl + iljl + i2j2)jl* + (i2 +J2 + i1j2 + i2jl)j2* = (i2 + J2 + i2j2 + i2jl) + [(i2* + i2*4I + il*j2) + j2* + (il* + ih*ji + i2*j2)j2* + (i2* + i2*jl + il*j2)jl*J + (i2 + J2 + i1j2 + i2jl)jl* + (il + jl + iljl + i2j2)j2* = (i2 + i2*) + (j2 + J2* + j2jl* + jlj2*) + (il + i1*) (j2 + i2* + j2jl* + jij2*) + (i2 + i2*)(ji +jl* +JllJ* +j2j2*)- Since j1 + jl* + jiljil* + 12j2* and j2 + j2* + jlj2* + j2j1* are in I + 1*, this equation implies that I + 1* is right semiregular. THEOREM 3. The sum R of all the right semiregular ideals of a semiring S is a right semiregutar two-sided ideal. Proof: Lemma 1 implies that R is right semiregular. If ri and r2 are in R, then rls and r2s are in R, for any s in S. Hence there exist elements r3 and r4 in R such that ris + r3 + r1sr3 + r2sr4 = r2s + r4 + rpsr4 + r2sr3. Whence, sr1sr, + sr3r, + sr1sr3r1 + sr2sr4r, = sr2sr, + sr4r, + sr1sr4r, + sr2sr3r, and sr2sr2 + sr4r2 + sr1sr4r2 + sr2sr,r2 = sr1sr2 + sr3sr2 + srisr3r2 + sr2sr4r2. Adding the last two equations and adding sr, + sr2 to both sides of the resulting equation we obtain that sr2 + (sr, + sr3r, + sr4r2) + sr1(sr1 + sr3r, + sr4r2) + sr2(sr4r, + sr2 + sr3r2) = sr, + (sr2 + sr3r2 + sr4rl) + sr2(sr3 + sr3r, + sr4r2) + sr,(sr2 + sr3r2 + sr4r,). Since s?1 + 166 MA THEMA TICS: S. BOURNE Proc. N. A. S. sr3r, + sr4r8 and sr2 + srar2 + sr4r, are in sR, this equation implies that sR is right semiregular and sR < R. Hence R is also a left ideal and R is a two-sided ideal. Definition 4: The right Jacobson radical R of a semiring S is the sum of all the right semiregular ideals of S. In a corresponding manner, we obtain the left Jacobson radical R' as the sum of all the left semiregular ideals of S. An ideal of S is said to be semiregular if it is both right semiregular and left semiregular. LEMMA 2. If i1 and i2 are contained in the right ideal I of S, and if iA + il + iljl + i2j2 = i2 +J2 + i2j2 + i2jiandil + k1 + kuii + k2i2 = i2+ k2 + k2i1 + kMi2, where ji, j2, k1 and k2 are in I, then there exists an element I in I such that ki + j2+ 1 = k2 + Jl + 1. Proof: ki + (iA + ji + iij1 + i2j2) + ki(il +jl + iljl + i2j2) + k,(i2 + j2 + i1j2 + i2jl) = ji + (i + k1 + k1ii + k2i2) + (i + k + kii + k2s)ji + (i2+ k2+ k2Si +kui2)j2. Hence k1 + (i2 + j2 + i1j2 + i2jl) + kl(i2 +J2 + i1j2 + i2jl) + k2(il + Jl + ilfl + i2j2) = ji + (i + k2 + k2ii + k1i2) + (i2 + k2 + k2i1 + kii2)jI + (iA + k1 + k1ii + k2ii)j2. Therefore, k1 + j2 + (ii + ki + k1ii + k24)j2 + (si + k2 + k1i2 + kgii)ji + i2 + k1i2 + k2iS = k2 + jl + (i2+ k2 ki+k+ + kji2)ju+ (i1+ k1 + kuii + k2i2) j2 + i + k2ii + k1i2. We define I = (i2 + k2 + k,il + kii)j + (iA + k1 + kii + kt4)j2 + it + kgil + kAi2. Then k++ j2+ I = k2 + j1 + 1, where I is in I. LEMMA 3. The right Jacobson radical R of a semiring S is a left semiregu- lar ideal. Proof: Theorem 3 states that R is a left ideal. We let r1 and r2 be any two elements of R. Since R is right semiregular, there exist elements si and st in R such that ri + si + r1s1 + r2s2 = r2 + s2 + ris2 + r2s,. Therefore, SI + t + s1t, + s2t2 = S2 + t2 + sits + s2ti, where ti and h are in R. By Lemma 2 there exists an element v in R such that t1 + r2 + v = t2 + r1 + v. Therefore, si + r1 + sit, + s2t2 + t4 + v = s2 + 4 + r1 + v + sit2 + s2t, or s1 + r1 + sit, + s2t2 + t1 + v = s2 + r2 + sit2 + s2t, + ti + v. Whence, si + ri + surn + s2t2 + sltl + slv + tl + v = s2 + r2 + si(ts + ri + v) + ti + v+ s2t1.or si + r1 + srl + s2t2 + sltl + SlV + tl + V = S2a + r2 + s2th + sltl + SIV + tl + V + s1r2. Finally, si + ri + surn + s2r2 + u = si + r2 + sirt + s2r, + u, where u is in R. Since si and s2 are in R, this equation implies that R is a left semi- regular ideal. THEOREM 4. The right Jacobson radical of a semiring S is equal tQ its left Jacobson radical R'. Proof: Lemma 3 implies that R < R'. From the symmetry of our results, we have that R' < R. Hence R = R'. VOL. 37, 1951 MA THEMA TICS: S. BOURNE 167

We may now refer to the Jacobson radical R of a semiring S. 4. Properties of the Jacobson Radical.-Definition 5: The semiring S is said to be semisimple if its Jacobson radical R = (0). Definition 6: The semiring S is said to be a radical semiring if its Jacob- son radical R = S. THEoREm 5. If R is the Jacobson radical of the semirsng S, then the dif- ference semiring S - R is semisimple. Proof: We let S = S - R and A be the Jacobson radical of S. If ri and r2 are in A, there exist elements si and S2 in A such that ri + S1 + fig, + 72S2 = f2 + S2 + r1S2 + f2S1. This implies that ri + si + ris, + r2s2 + r3 = r2 + S2 + r1s2 + r2s1 + r4, where r3 and r4 are in R, and rl, r2, s, and 52 are in the set R* of all elements of S belonging to the cosets of A. R* is an ideal in S, since A is an ideal in S. R* contains R, since 0 is in A. Since r8 and r4 are in R, there exist elements sJ and S4 in R such that r4 + S4 + r4s4 + r3s3 = r3 + s3 + r3s4 + r4s3. If we denote the left-hand side of this equation by (e) and its right-hand side by (e)', then (e) + s3 + (e)s4 + (e)'ss = (e)' + S3 + (e)'s4 + (e)s3 yields that r1 + si + ris1 + r2s2 + r7 + s5 + r1s4 + S1S4 + rl1ss4 + r2s2s4 + rs4 + r2s3 + s2s3 + r1s2s, + r2s1s, + r4s3 = ri + si + s1s4 + s2s3 + ri(s, + s4 + s1s4 + s2s3) + r2(s2 + s2s4 + s3 + siss) + r3 + s5 + r3s4 + r4s3 = r2 + s2 + S2s4 + s5s5 + rj(s2 + S3 + S2S4 + S1sO) + r2(s1 + S4 + SS4 + S25O) + r4 + S3 + r4s4 + ras3 We let ti = si + sis4 + s2s3, t2 = s2 + s2s4 + sis3 t8 = r, + rs54 + r7s4 14 = r4 + r4s4 + r&s3. Since r4 + S4 + r4s4 + r3S3 = r8 + sJ + r3s4 + r4ss, we have that t3 + ss = t4 + s4. On substituting we obtain that ri + t1 + ri(t, + 54) + r2(12 + sQ) + 14 + S4 = r2 + t2 + ri(12 + 58) + r2(t, + 54) + 4 + ss. On adding to both sides (r1 + r2)t4 we obtain that r1 + (t1 + s4 + tQ) + r1(t1 + S4 + 4) + r2(t2 + S3 + t4) = r2 + (t2 + S8 + t4) + r(t2 +- S3 + 4) + r2(ti + S4 + t4). Since 11 + S4 + 4 and 12 + S3 + 14 are in R*, this equation states that R* is a semiregular ideal in S. Therefore R* < R and iA = (0). THEoREM 6. The Jacobson radical of a semsring S is a radical semirsng. Proof: We let R(S) denote the Jacobson radical of S and R(R(S)) the Jacobson radical of R(S). Since R(S) is semiregular in R(S), then R(S) < R(R(S)). Trivially, R(R(S)) < R(S). Whence, R(R(S)) =- R(S). Thus the structure of an arbitrary semiring is reduced to the considera- tion of semisimple semirings and radical semirings. I j If r' = 0, then on-letting r' = r2'andr'r = r2i - we obtain that r + r' + rr' = r' + rr'. Hence if an element is in S, it is automatically semiregular. However, the converse of this statement is 168 MA THEMA TICS: S. BOURNE PROC. N. A. S. false. For, in the semiring of non-negative rational numbers 2 is semi- regular while not nilpotent.6 THEOREM 7. Every nilpotent right or left ideal of a semiring S is contained in the Jacobson radical R of S. Proof: We let I be a nilpotent right ideal of degree n, that is In = (0). We wish to prove that I is a right semiregular ideal. It can be easily veri- fied that, if we define 2n -1

St = et(ji, *,ji.)ij,* . ik t = 1, 2 k-1 ji, ..., jk- 1, 2 and 1 if ji = 1 for an even of s, ( * . ,jk) = S k, Oifj, = 1 for an odd number of s, 62 = 1 el(jl, *k) then i1 + si + ils + i2s2 = i2 + S2 + i2S, + i1s2 where si and s2 are in I. This implies that I is a right semiregular ideal and that I < R. Definition 7: The principal right ideal (v) is the set of elements {vs + vn }, where s is an arbitrary element of the semiring S and n a non-negative . COROLLARY. If v is an element such that SvS < R, then v < R. Proof: If I is the principal right ideal (v), then I8 < SIS. However, SvS = SIS. Trivially, SvS < SIS. An element of SIS is of the form ESl(VS + vn)S2 = ESivSs2 + sivns2 = Esi'vs2'. Therefore, SIS < SvS and the result follows. Thus P < SvS < R. Now (I+ R)$ = E2(ii + ri) (i2 + r2) (8 + r3) = i2ii + Er. Therefore, (I + R)8 < R. We have then that I = (I + R) - R is a nilpoint ideal of degree 3 in the semisimple semiring = S - R. Theorem 7 states that I < A = (0), by theorem 6. Therefore I < R, and, in particular, v e R. Definition 8: A semiring S is said to be regular, in the sense of von Neumann,7 if for every element s in S there exist elements x' and x' in S such that sx's = s + sx's. We notice that in the case S is a ring, this definition reduces to the one given by von Neumann. THEOREM 8. If R is the radical of a regular semiring S, then for any ele- ment r in R there exists an element w in R such that r + w = w. Proof: We let r e R, then rx'r = r + rx'r. Since x'-' and x'r are in R, there exist elements si and S2 in R such that x"r + si + (X r)s, + (x'r)s2= x'r + s2 + (x'r)si + (x"r)s2. Therefore, rx'r + rs, + rx'rs, + rx'rs2 = rx r + rs2 + rx'rsi + rx'rs2. Whence, rx"'r + rs, + rx"rs, + rs2 + rx'rs2=r + rx'r + rs2+ rs +rx"rs, + rx"rs2. We define w = rx'r + rs, + rx'rs, + rs2 + rx'rs2 and obtain that r + w = w, w e R. VOL. 37, 1951 MA THEMA TICS: S. BOURNE 169

We notice that in the case S is a regular ring, this theorem implies that S is a semisimple ring. 5. The Jacobson Radical of a Matrix Semiring Sn. LEMM 4. If si, is right semiregular in the semiring S, then the matrix M = (sjj), sij = 0, i > 1, is right semiregular in the matrix semiring S,. Proof: Theorem 2 yields that s + s' + slis' = s' + slis'. Therefore, Sij + slj' + sllslj' = slj' + sllsl5j. We have, then, that M + M' + MM' = M' + MM", where M' = (Sij'), Sij' = 0, i > 1, and M' = (si), sjj' = 0, i > 1. Thus M is right semiregular in S,. LEMMA 5. If R is the Jacobson radical of a semiring S, then the right ideal I ofall matrices M = (rjj), rjj e R, rjj = O, i # k, is a right semiregular ideal in Sn. 0 Proof: The fact that Ik is a right ideal is obvious. We let M1 = (rjj) and M1* = (rtj*) be any two matrices of Ik. We consider the pairs of elements (rkj, rkj*), j = 1, 2, ..., n. Since rkj and rkj* are in R, there exist elements sk, Skj in R such that rkj + Skj + rksjSk + rkj*Skj = rj* + skj* + rkj*Skj + rkjskj* We define S, = (s5j), sj = 0, i $ k and S,* = (sij*), sjj* = 0, i 0 k. Then, we can easily verify that M1 + S, + M1Sl + Ml*S,* = M1* + S1,* + Ml*S, + M1Sl*, where S, and S,* are in Ik. We have proved that Ik is a right semiregular ideal in Sn. THEOREM 9. If R is the Jacobson radical of a semiring S with a element, then Rn is the Jacobson radical R(Sn) of the matrix semiring Sn. Proof: Now Rn = Ik. Since Lemma 5 states that Is is a right semi- regular ideal, then Rn is a right semiregular ideal by Lemma 1. Hence, Rn < R(Sn). We wish to prove that R(Sn). < Rn. We let M = (sjj)e R(Sn) and Ttf = (tim),tim = O I ,li,m $j, tij 1. Then D= [s,, ...,Spq] = EkTkVMT.t is contained in R(Sn). We consider the totality of diagonal matrices of the form D = [d, ..., dJ which are contained in R(Sn). Then the set A of elements, which are contained in the first place of D, forms a right ideal in S. If D1 = [d4, di] and D2 = [d2, ..., d4] are in R(Sn), there exist matrices S, and S2 in R(Sn) such that D1 + S, + DIS1 + D2S2 = D2 + S2 + D1S2 + D2S1. Using matrices of type Tij, previously defined, we can modify this to read D1 + E1 + DIE, + D2E2 = D2 + E2 + D1E2 + D2E1, where Ek = [ellk, ellk, .., ellk], k = 1, 2. We have then that di + ell' + die,,' + d2e,12 = d2 + ell2 + die12 + d2e1ll, where ell' and ell2 are in A. This implies that A is a right semiregular ideal and A < R. This yields that s,,, e R and M < Rn. Therefore, R(Sn) . Rn and R(Sn) = Rn-

* Present address: University of Connecticut, Storrs, Conn. 'Vandiver, H. S., "Note on a Simple Type of in Which the Cancellation Law of Addition Does Not Hold," Bull. Am. Math. Soc., 40, 920 (1934). 170 MA THEMA-TICS: HARISH-CHANDRA PROC. N. A. S.

'Vandiver, H. S., "On Some Simple Tpes of Semi-rings," Am. Math. Monxhly, 46,22- 26 (1939). 'Jacobson, N., Theory of Rings, New York, 1943, p. 2. 4Jacobson, N., "The Radical and Semi-simplicity for Arbitrary Rings," Am. J. Math., 67, 300-320 (1945). 'Dubreil, P., "Contribution a la theorie des demi-groupes," Mem. Acad. Sci. Inst. France (2), 63, No. 3, 52 (1941). McCoy, N. H., Rings and Ideals, Math. Assoc. Am., Buffalo, 1948, p. 137. von Neumann, J., "On Regular Rings," PROC. NATL. ACAD. Sci., 22, 708 (1936).

REPRESENTATIONS OF SEMISIMPLE LIE GROUPS ON A BANACH BY HARIsH-CEIANDRA DEPARTMENT OF MATHETICS, COLUMBIA UNIVBRSTrr Communicated by 0. Zariski, January 26, 1951 Let G be a connected semisimple Lie and D a Banach space. By a representation of G on i we mean a mapping -r which assigns to every x e G a bounded linear operator r(x) on i such that the following two con- ditions are fulfilled: (1) -r(xy) = zr(x)ir(y) and wr(l) = I (here 1 is the unit element of G and I the identity operator on 0. (2) The mapping (x, A) -- r(x)#(x e G, ^,6 e ) is a continuous mapping of G X j into . The object of this note is to announce a few theorems on these representa- tions. No attempt is made to give proofs here. A detailed account with complete proofs will appear elsewhere in another paper. Let R and C, respectively, be the fields of real and complex numbers. Let g0 be the of G and g the complexification of go. We denote the universal enveloping' algebra of g by !5. Let CcX(G) be the set of all complex-valued functions on G which are indefinitely differentiable every- where and which vanish outside a compact set. Let r be a representation of G on 0 and V the set of all elements in 0 which can be written as finite linear combinations of elements of the form fGf(x)r(x) dx (' e I, f e Cc (G)) where dx is the left invariant Haar measure on G. V is called the Girding subspace2 of 0 (with respect to z-). For every X e g0 we can define a linear transformation 7r(X) of V into itself such that Tv(x)41 = lim -{w(exp tX) -I}iP ( e V, t e R). t-0g o t