462 ERRATA PROC. N. A. S. ERRA TA Dr. R. E. Johnson has called to my attention that Theorem 1 of my paper "The Jacobson Radical of a Semiring" in these PROCEEDINGS, 37, 163-170 (1951) is in part incorrect. This theorem should read as follows: If I is an ideal of a semiring S, then S is homomorphic to the difference semiring S = S - I. SAMUEL BOURNE Downloaded by guest on September 30, 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 numbers. 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 isomorphisms 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 polynomial in n variables, we see that the n' iso- morphisms 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 lattice 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 set S together with two binary operations, called addition and multiplication, which forms a semigroup 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 semirings 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 endomorphisms 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 homomorphism 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 matrix 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 zero element. 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 modulo 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 coset of the ideal I. Relative to the usual definitions of an addition and multiplication the cosets of an ideal I of a semiring form a semiring, called the difference semiring S - I. The correspondence s -' s defines a homo- morphism 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.