1_dutta.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No. 4, 2012, 401-413 ISSN 1307-5543 – www.ejpam.com Weakly Special Radical Class and Special Radical Class of Ternary Semirings Tapan K. Dutta1, Kar Ping Shum 2,∗, Shobhan Mandal1 1 Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata- 710019,India 2 Institute of Mathematics, Yunnan University, Kunming, 650091, People Republic of China Abstract. In this paper, we consider the weakly special radical classes and special radical classes of ternary semirings. Some of our results are similar to those in rings theory as well as in semiring theory. In particular, the upper radicals of the above two classes are determined. 2010 Mathematics Subject Classifications: 16Y60, 16Y99 Key Words and Phrases: weakly special radical classes, special radical classes, singular radicals, spe- cial singular radical of ternary semirings. 1. Introduction In this paper, we consider “The Singular Ideal of a ternary Semiring” mentioned in [3]. The notion of a ternary semiring was first introduced by T. K. Dutta and S. Kar in [3]. Subse- quently, many related notions of semiring and ring have been generalized to ternary semirings. Some earlier works of ternary semiring may be found in [4]-[12] and [13, 14, 15, 16]. The Partitioning and subtractive ideals of ternary semirings were considered by J. N. Chaudhari and K. J. Ingale in [2]. For the general radical theory of rings,the reader is referred to the classical monograph of N. J. Divinsky [17]. For definitions and properties of ideals, homomor- phism, quotient for ternary semirings,singular ideals, singular ternary semirings, non-singular ternary semiring, the reader is referred to [2]. The concepts of radical class for hemirings were given by D. M. Olson and T. L. Jenkins in 1983, see [27]. Moreover,the general theory to upper radicals was extended by A. C. Nance in [27] and the special radical classes and properties of special radicals were investigated by M. D. Olson, G.A.P. Heyman and H. J. L. Roux in [26]. The properties of the weakly special radical class of hemirings were also studied by the above authors. ∗Corresponding author. Email addresses: duttatapankumar�yahoo. o.in (T. Dutta), kpshum�ynu.edu. n (K. Shum), shobhan139�gmail. om (S. Mandal) http://www.ejpam.com 401 c© 2012 EJPAM All rights reserved. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 402 In this paper, we first extend the above notions to ternary semirings. Then, we define again the weakly special radical class and special radical class for ternary semirings. We will show that the class of all semiprime and non singular ternary semirings forms a weakly spe- cial radical class whereas the class of all prime non-singular ternary semirings forms a special radical class. As a consequence of this result, the upper radicals will be determined by the above two classes which are called the singular radical and special singular radical of ternary semirings, respectively. For the radical properties,the reader is referred to the well known monograph of N. J. Divinsky. Throughout this paper, S will be used to denote a ternary semiring with zero and S∗ = S \ {0}. Also we use M to denote a right ternary S-semimodule with zero. 2. The Radical Class, Weakly Special Radical Class and Special Radical Class Let S be a ternary semiring and M a right ternary S-semimodule. We define ZS(M) [2] by {m ∈ M : rS(m) be an essential right ideal of S}. We first give the following crucial definition. Definition 1 ([2]). A ternary subsemimodule ZS(M) of M is called a singular ternary subsemi- module of the right ternary S-semimodule M. We call a singular ternary subsemimodule ZS(S) a right ideal of a ternary semiring S and call this kind of right ideal the (right) singular ideal [2] of a ternary semiring S which is denoted by Z(S), that is, Z(S) = {t ∈ S : rS(t) is an essential right ideal of S} . A ternary semiring S is said to satisfy the condition α[2] if for any nonzero element a in S, rS(a) 6= S or equivalently aSS = 0 implies that a = 0. The properties of singular modules of a ternary modules are given in the following propo- sitions, see [2]. Proposition 1. ZS(M) is a ternary subsemimodule of M. Proposition 2. Let S be a ternary semiring with condition α. Then the singular ideal Z(S) is a k-ideal of S. Proposition 3. If I is an ideal of a ternary semiring S and as a ternary semiring, I is semiprime, then Z(I) = I ∩ Z(S). We give the following example to show that there exists a prime(semiprime) ternary semir- ing satisfying the condition α. Example 1 ([2]). Let S be a prime(semiprime) ternary semiring. Then S satisfies the condition α. The following theorem of ternary semirings can be found in [2]. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 403 Theorem 1. Let S and S′ be two semi-isomorphic ternary semirings. Then S is singular(nonsingular) if and only if S′ is singular(resp. nonsingular). We state the following known definition. Definition 2 ([5]). A non-empty subset A of a ternary semiring S is called a p-system if for each a ∈ A there exist elements x1, x2, x3, x4 of S such that ax1ax2a ∈ A or ax1 x2ax3 x4a ∈ A or ax1 x2ax3ax4 ∈ A or x1ax2ax3 x4a ∈ A. In the following theorem, we characterize the semiprime ideal of a ternary semiring. Theorem 2 ([5]). A proper ideal Q of a ternary semiring S is semiprime if and only if its complement P c is a p-system. Proposition 4. Let S be a ternary semiring. If Q is a semiprime ideal of S and I is an ideal of S then Q ∩ I is a semiprime ideal of I. Proof. Let J be an ideal of I such that J3 ⊆ I ∩Q. Then J3 ⊆ Q. If possible, let J 6⊆ I ∩Q. Then J 6⊆ Q. Hence, there exists an element a ∈ J but a 6∈ Q. Now by theorem 2, Q is a p-system. Then a ∈ Qc implies that there exist elements x1, x2, x3, x4 of S such that ax1ax2a ∈ Qc or ax1 x2ax3 x4a ∈ Qc or ax1 x2ax3ax4 ∈ Qc or x1ax2ax3 x4a ∈ Qc. We con- sider the following situation: If ax1ax2a ∈Qc, then there exist elements s1, s2, s3, s4 of S such that ax1ax2as1ax1ax2as2ax1ax2a ∈Qc or ax1ax2as1s2ax1ax2as3s4ax1ax2a ∈Qc or ax1ax2as1s2ax1ax2as3ax1ax2as4 ∈ Qc or s1ax1ax2as2ax1ax2as3s4ax1ax2a ∈Qc. Now consider the following cases: (i) ax1ax2as1ax1ax2as2ax1ax2a =a(x1ax2as1ax1ax2as2)a(x1ax2)a =ai1ai2a ∈ J3 ⊆ Q as i1, i2 ∈ I where i1 = x1ax2as1ax1ax2as2 and i2 = x1ax2. (ii) ax1ax2as1s2ax1ax2as3s4ax1ax2a =a(x1ax2as1s2ax1ax2as3s4)a(x1ax2)a =ai1ai2a ∈ J3 ⊆ Q as i1, i2 ∈ I where i1 = x1ax2as1s2ax1ax2as3s4 and i2 = x1ax2. (iii) ax1ax2as1s2ax1ax2as3ax1ax2as4 =a(x1ax2)a(s1s2a)(x1ax2as3ax1)a(x2as4) =ai1ai2i3ai4 ∈ J3 ⊆ Q as i1, i2, i3, i4 ∈ I where i1 = x1ax2, i2 = s1s2a, i3 = x1ax2as3ax1 and i4 = x2as4. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 404 (iv) s1ax1ax2as2ax1ax2as3s4ax1ax2a =(s1ax1)a(x2as2)a(x1ax2)(as3s4ax1ax2)a =i1ai2ai3i4a ∈ J3 ⊆ Q as i1, i2, i3, i4 ∈ I where i1 = s1ax1, i2 = x2as2, i3 = x1ax2 and i4 = as3s4ax1ax2. Now, from (i) , (ii), (iii) and (iv), we can easily see that ax1ax2a 6∈Qc. Hence, J ⊆ Q, and whence, J ⊆ I ∩Q. This proves that I ∩Q is a semiprime ideal of S. We state below the following definition of m-system of a ternary semiring. Definition 3 ([4]). A non-empty subset A of a ternary semiring S is called an m-system if for each a, b, c ∈ A there exist elements x1, x2, x3, x4 of S such that ax1 bx2c ∈ A or ax1 x2 bx3 x4c ∈ A or ax1 x2 bx3cx4 ∈ A or x1ax2 bx3 x4c ∈ A. We now characterize the prime ideals of a ternary semiring. Theorem 3 ([4]). A proper ideal P of a ternary semiring S is prime if and only if its complement P c is an m-system. Proposition 5. Let S be a ternary semiring and P a prime ideal of S. If I is an ideal of S. Then P ∩ I is a prime ideal of I. Proof. Proceeding as in Proposition 4, the proposition follows immediately. In below, we let S = {S : S be a ternary semiring such that every nonzero homomorphic image S′ of S contains a nonzero ideal which is singular as a ternary semiring}. Proposition 6. S = {S : S is a ternary semiring such that for every nonzero homomorphic image S′ of S, β(S′) 6= 0 or Z(S′) 6= 0}. Proof. Let S ′ = {S : S is a ternary semiring such that for every nonzero homomorphic image S′ of S,β(S′) 6= 0 or Z(S′) 6= 0}. Let S ∈ S and S′ be a nonzero homomorphic image of S. So S′ contains a nonzero ideal I (say) which is singular as a ternary semiring i.e. Z(I) = I . Suppose β(S′) = 0. Then S′ is a semiprime ternary semiring; so I is a semiprime ternary semiring by Proposition 4. Hence,by Proposition 3, Z(I) = I ∩ Z(S′) i.e. I ∩ Z(S′) = I which implies that I ⊆ Z(S′). Consequently Z(S′) 6= 0. Hence β(S′) 6= 0 or Z(S′) 6= 0. Therefore, S ∈ S ′. Conversely, let S ∈ S ′ and S′ be a nonzero homomorphic image of S such that β(S′) 6= 0 or Z(S′) 6= 0. Suppose that β(S′) 6= 0. Then S′ contains a nonzero ideal I such that I3 = 0. We now prove that Z(I) = I . For this purpose,let x ∈ I and H a nonzero right ideal of I . Then xHI ⊆ x I I ⊆ I I I = 0. This leads H ⊆ rI(x),that is, rI (x)∩ H = H 6= 0. Therefore, x ∈ Z(I). Hence, Z(I) = I . If β(S′) = 0, then Z(S′) 6= 0. Since β(S′) = 0, S′ is a semiprime ternary semiring. Let Z(S′) = I . Then, by Example 1 and Proposition 2, I is a nonzero ideal of the T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 405 semiprime ternary semiring S′. Hence. by Proposition 4, I is a semiprime ternary semiring. Thus, by Prop 3, Z(I) = I ∩ Z(S′) = I that is,I is singular as a ternary semiring. In both cases S ∈ S . Thus, S = S ′. Recall the following definition of hereditary class given in N. J. Divinsky [3]. Definition 4. A class ρ of ternary semirings is called hereditary if I is an ideal of a ternary semiring S and S ∈ ρ then I ∈ ρ. Now, by D. M. Olson and A. C. Nance [27], we define the regular class in a ternary semiring as follows: Definition 5. A classM of ternary semirings is called regular if S ∈M and I is a nonzero ideal of the ternary semiring S, then there is a nonzero homomorphic image of I inM . Following D. M. Olson and A. C. Nance [27], we define the radical class in ternary semir- ing. Definition 6. A nonempty class γ of ternary semirings is called a radical class if the following conditions hold: (R1) γ is homomorphically closed. (R2) If S 6∈ γ, then S contains a proper k-ideal K such that S/K has no nonzero γ-ideals (ideals which are as ternary semirings are in the class γ). As an example of the radical class in ternary semiring, we have the following Lemma. Lemma 1. If I is a nil ideal of a ternary semiring S, then I is also a nil ideal of S. Proof. Let I be a nil ideal of S and x ∈ I . Then x + i ∈ I for some i ∈ I Since I is nil, for each t in S there exists a positive integer n(depending on t) such that [(x+ i)t]n(x+ i) = 0. Since I is an ideal of S, so [(x+ i)t]n(x+ i) can be written as (x t)n x+h for a particular h ∈ I . Thus, (x t)n x+h= 0, and as a result, we have (x t)n+1h+hth= 0. Now, (x t)2n+1 x = (x t)n+1[(x t)n x + h] + hth = hth. But, hth ∈ I , so (x t)2n+1 x ∈ I . (x t)2n+1 x is nilpotent. Hence, for each t ∈ S, [(x t)2n+1 x t]k[(x t)2n+1 x] = 0 for some positive integer k ⇒ (x t)(2n+2)k+2n+1 x = 0. Thus, x is nilpotent and thus I is a nil ideal of S. Theorem 4. The class N of all nil ternary semirings is a radical class. Proof. Clearly, the class N is homomorphically closed. Let S be a ternary semiring such that S 6∈ N . Now, using Zorn’s Lemma, we choose an ideal M of S which is maximal with respect to being a nil ideal. Since S 6∈ N , M is a proper ideal of S. By Lemma 1 and the maximality of M , we see that M = M . This means that M is a k-ideal of S. If I/M is any N -ideal of S/M then for any x ∈ I , x/M is nilpotent. Now, for each t/M ∈ S/M , there exists a positive integer n such that [(x/M)(t/M)]n x/M = ((x t)n x)/M = 0/M . But then (x t)n x ∈ M which makes (x t)n x and therefore, x is nilpotent. Thus, I is a nil ideal of S, and so I ⊆ M since M is maximal. Hence, I/M = (0), and so S/M has a no non-zero N -ideal. This shows that N is a radical class. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 406 Lemma 2. If φ is a semi-isomorphism from a ternary semiring S onto a ternary semiring T and I is a nonzero ideal of S, then φ(I) is a nonzero ideal of T . Proof. Clearly, φ(I) is an ideal of T . If φ(I) = (0), then, I ⊆ Kerφ = (0), as φ is semi- isomorphism. Thus I = (0), a contradiction. This shows that φ(I) is a nonzero ideal of T . The following Theorem is a theorem for the regular radical class of the ternary semirings. Theorem 5. IfM is a regular class of ternary semirings, thenUM = {ternary semirings S : no nonzero homomorphic image of S is inM} is a radical class. Proof. Suppose that S ∈ UM and φ(S) is a nonzero homomorphic image of S. Let ψ(φ(S)) be a nonzero homomorphic image of φ(S). Then ψ(φ(S)) = (ψφ)(S) is a nonzero homomorphic image of S. Since S ∈ UM , (ψφ)(S) 6∈ M . Hence φ(S) ∈ UM . Thus UM is homomorphically closed. Next, we suppose that S 6∈ UM . Then there exists a nonzero homomorphic image φ(S) ∈M . Now as φ is nonzero, Kerφ is a proper k-ideal of S and S/Kerφ ≃ φ(S), and let the semi-isomorphism beψ. If I is a nonzeroUM -ideal of S/Kerφ, then by Lemma 2,ψ(I) is a nonzero ideal of φ(S). Since I ∈ UM and UM is homomorphically closed, ψ(I) ∈ UM . Since φ(S) ∈ M and M is regular, ψ(I) has a nonzero homomorphic image in M . This, however, contradicts toψ(I) ∈ UM , As a result, we have shown that S/Kerφ has no nonzero UM -ideals.Thus, UM is indeed a radical class. Definition 7. Let S be a ternary semiring and A be a nonempty subset of S. Then, the annihilator of A in S, denoted by annS(A), is defined by {x ∈ S : Axs = 0 and Asx = 0 for all s ∈ S}. Proposition 7. Let S be a ternary semiring and A be a right ideal of S. Then annS(A) is a k-ideal of S. Proof. Obviously, annS(A) is nonempty since 0 ∈ annS(A). Also, if a, b ∈ annS(A), then a+ b ∈ annS(A). Let x ∈ annS(A) and s1, s2 ∈ S. Then, Axs = 0 and Asx = 0 for all s ∈ S and so Axs1s2s = 0 and Asxs1s2 = 0 for all s ∈ S. This leads to xs1s2 ∈ annS(A). Hence, annS(A) is a right ideal of S. Also As1s2 xs ⊆ Axs = 0 and Ass1s2 x ⊆ ASx = 0 for all s ∈ S as A is a right ideal of S. So s1s2 x ∈ annS(A). Hence, annS(A) is a left ideal of S. Again, we have As1 xs2s = (As1 x)s2s = 0 and Ass1 xs2 ⊆ AxS = 0 for all s ∈ S. So s1 xs2 ∈ annS(A). Hence, annS(A) is a lateral ideal of S. Thus, annS(A) is an ideal of S. Now let a, a + b ∈ annS(A). Then Aas = 0 = A(a+ b)s and Asa = 0 = As(a + b) for all s ∈ S. This implies Abs = 0 and Asb = 0 for all s ∈ S. So, b ∈ annS(A). This proves that annS(A) is a k-ideal of S. Following D. M. Olson, G.A.P. Heyman and H. J. L. Roux [26], we define he weakly special radical class in ternary semirings as follows: Definition 8. A classM of ternary semirings is called a weakly special radical class ifM is a hereditary class of semiprime ternary semirings satisfying the following conditions: (x) If S is a ternary semiring and S is semi-isomorphic to T with T ∈M , then S ∈M . T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 407 (z) If I ∈M and I is an ideal of a ternary semiring S, then S/annS(I) ∈M . We give the following crucial Lemma. It is noted that the h-prime and h-semiprime ideals in semirings and Γ-semirings have been recently investigated by S. Sardar and others in [29]. For semiprime ternary semirings, we have the following main theorem. Theorem 6. If M is a hereditary class of semiprime ternary semirings which satisfies proper- ties: “if S ∈ M and S is semi-isomorphic to T then T ∈ M ” then the following conditions are equivalent: (1) If A∈M and A is an ideal of S, then S/annS(A) ∈M ; (2) If A∈M with A an ideal of S and annS(A) = 0, then S ∈M . (3) If A∈M and A is an essential ideal of S, then S ∈M . Proof. For an hereditary radical class of semiprime ternary semirings, it is clear that (1) ⇒ (2). For (2) ⇒ (3), suppose that A is an essential ideal of S and A ∈ M . Now let a ∈ A∩ annS(A), then AaS = 0 and ASa = 0. This implies that aAa = 0 and AAa = 0 since a ∈ A and A⊆ S. Thus aAaAa = 0, aAAaAAa = 0, aAAaAaA= 0 and AaAaAAa = 0.Thus, a = 0 since A is semiprime. Because A is essential, annS(A) = 0. Now by (2), S ∈ M . Assume that (3) holds and consider a ternary semiring A ∈M with A an ideal of S. Then, by using the arguments as in above, we see that A∩ annS(A) = (0) since A is semiprime. Now (A+ annS(A))/annS(A) is a nonzero ideal of S/annS(A). Also, we have A= A/(A∩ annS(A))≃ (A+ annS(A))/annS(A), and so by the given condition,we deduce that (A+ annS(A))/annS(A) ∈ M . However,(A+ annS(A))/annS(A) is an essential ideal in S/annS(A),for if H/annS(A) is a nonzero ideal of S/annS(A), then H ∩ A 6= 0, otherwise, AHS,ASH ⊆ H ∩ A would be zero which implies that H ⊆ annS(A) which is impossible. Also, (H∩A)∩annS(A) = (0). Thus, there exits an a( 6= 0) ∈ H∩A such that a/annS(A) 6= 0/annS(A). Also, a/annS(A) ∈ [A+ annS(A)/annS(A)]. Hence, [A+ annS(A)/annS(A)] ⋂ (H/annS(A)) 6= 0/annS(A). But then, by the condition (3), we have S/annS(A) ∈M and so, condition (1) follows, as desired. Lemma 3. If the ternary semirings S ≃ T and T are semiprime, then S is semiprime. Proof. Suppose that φ is a semi-isomorphism and A is an ideal of S such that A3 = (0). Then, φ(A) is an ideal of T and φ(A3) = (0)⇒ [φ(A)]3 = (0). Thus, φ(A) = (0) since T is semiprime⇒ A⊆ Kerφ = (0)⇒ A= (0). This shows that S is a semiprime ternary semiring. The semiprime ternary semirings is described in the following Lemma. Lemma 4. If the ternary semirings S ≃ T and S are semiprime, then T is semiprime. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 408 Proof. Suppose that φ : S → T is a semi-isomorphism and A′ is an ideal of T such that A′3 = (0). Then, there exists an ideal A of S such that φ(A) = A′, as φ is surjective. Now φ(A3) = [φ(A)]3 = A′3 = (0). Thus A3 = (0), as kerφ = 0. Since S is semiprime, A= (0), we have A′ = (0). Thus, T is a semiprime ternary semiring. In the following theorem, we consider the semiprime non-singular ternary semirings. Theorem 7. The class ℘ of semiprime non-singular ternary semirings forms a weakly special radical class. Proof. Let S ∈ ℘ and I be a nonzero ideal of S. Then by Lemma 4, I is a semiprime ternary semiring. Also, from Proposition 3, Z(I) = I ∩ Z(S) = (0), since Z(S)=0 and S is nonsingular, I ∈ ℘. Thus, the class ℘ is a hereditary class of semiprime nonsingular ternary semirings. By Theorem 1 and Lemma 3, the class ℘ satisfies the property (x) of the definition of weakly special radical class. In order to prove the condition (z) of the weakly special radical class, in view of Theorem 1, Lemma 4 and Lemma 6, it suffices to prove that if I ∈ ℘ and I is an essential ideal of a ternary semiring S then S ∈ ℘. To prove that S is a semiprime ternary semiring, we let K3 = 0, where K is an ideal of S. Let K ′ = K ∩ I . Then K ′ is an ideal of I . Now K ′3 ⊆ K3 = 0 implies K ′ = 0 since I is a semiprime ternary semiring. Since I is an essential ideal and so K = 0. This shows that S is a semiprime ternary semiring. Again if S is not a nonsingular ternary semiring, then by Example 1 and Proposition 2, we can easily see that Z(S) is a nonzero ideal of S and thus I ∩ Z(S) 6= 0, since I is an essential ideal of S. Hence, by Proposition 3, Z(I) 6= 0 which is a contradiction, since I ∈ ℘. Therefore, S is nonsingular and hence, S ∈ ℘. This shows that ℘ is a weakly special radical class. 3. Supernilpotent Radical Class and Weakly Special Radical Class of Ternary Semirings We first give the following useful Definition. Definition 9. Let S be a ternary semiring. Then, we call an ideal I of S nilpotent if there exists a positive integer n such that I2n+1 = 0. The semiring S is said to be a nilpotent ternary semiring if S is nilpotent as an ideal of itself. Following D. M. Olson and A. C. Nance [27], we define the supernilpotent radical class in ternary semiring as follows: Definition 10. A radical class of ternary semirings is called a supernilpotent radical class if it is hereditary and contains all the nilpotent ternary semirings. The following Lemma is a crucial lemma. Lemma 5. Let S be a ternary semiring. If I is a semiprime k-ideal J and J is an ideal of S, then I is an ideal of S. T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 409 Proof. Since I is a semiprime k-ideal of J , it is easy to see that J/I is a semiprime ternary semiring. Now, (I+SIS+JSISJ) is an ideal of J and we have (I+SIS+JSISJ/I)5 ⊆ I/I = 0/I . Thus, I + SIS + JSISJ/I = 0/I as J/I is semiprime ternary semiring. Since I is k-ideal of J ,SIS ⊆ I . Again, (I + ISS) is an ideal of J and (I + ISS/I)3/I ⊆ I/I = (0). Thus, (I + ISS)/I = 0/I as J/I is a semiprime ternary semiring. Now as I is a k-ideal of J , ISS ⊆ I . Also, (I+SSI) is an ideal of J and (I+SSI/I)3 ⊆ I/I = (0). Hence,we have (I+SSI)/I = (0) as J/I is a semiprime ternary semiring. Now as I is a k-ideal of J , we have SSI ⊆ I . This proves that I is an ideal of S. Corollary 1. Let S be a ternary semiring. If I is a prime k-ideal of J and J is an ideal of S,then I is an ideal of S. We now formulate a theorem of weakly special radical class of ternary semirings. Theorem 8. IfM is a weakly special radical class of ternary semirings, then UM = { ternary semirings S : no nonzero homomorphic image of S is inM} is a supernilpo- tent radical class. Proof. SinceM is a weakly special radical class of ternary semirings, M is a hereditary class. Let S ∈M and I be a nonzero ideal of S. Then I ∈M . Now I is a homomorphic image of itself. This means that M is regular. Hence, by Theorem 5, UM is a radical class. In order to show that UM is a hereditary class , let S ∈ UM and J be a nonzero ideal of S. If J 6∈ UM , then there is a nonzero homomorphic image φ(J) of J inM . Let K = kerφ. Then K is a k-ideal of J and J/K ≃ φ(J). Since φ(J) ∈M , φ(J) is semiprime, and so by Lemma 3, J/K is semiprime. Now φ(J) is nonzero, and hence, J/K 6= (0). Since K is a k-ideal, K is a semiprime ideal of J and, whence K is an ideal of S, by Proposition 5. Now J/K is a nonzero ideal of S/K , and so by the property (z), (S/K)/annS(J/K) ∈ M , since by the property (x), J/K ∈ M . But, if S/K ⊆ annS(J/K), then we have (J/K)3 = (0) which cannot happen since J/K is a semiprime ternary semiring. This leads to annS(J/K) 6= S/K . Thus, (S/K)/annS(J/K) is a homomorphic image of S which is inM and (S/K)/annS(J/K) 6= (0), since any annihilator ideal is necessarily a k-ideal. However,this is impossible because S ∈ UM . Hence, we deduce that J ∈ UM and hence, we have proved that UM is hereditary. Finally, if S is any nilpotent ternary semiring, then φ(S) is nilpotent for any nonzero homomorphism φ, and hence φ(S) 6∈ M . Thus, S ∈ UM and hence,UM is supernilpotent. We here call UM the upper radical class determined by the classM . Proposition 8. S = {S : S is a ternary semiring such that every nonzero homomorphic image S′ of S contains a nonzero ideal which is singular as a ternary semiring} is an upper radical class determined by the class ℘ of semiprime non-singular ternary semirings. Proof. Now U℘ = {S : no nonzero homomorphic image of S is in ℘} = {S : for every nonzero homomorphic image S1 of S either β(S1) 6= 0 or Z(S1) 6= 0} = S (by Proposition 6, since S1 ∈ ℘ implies β(S1) = 0 and Z(S1) = 0). T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 410 We now simply call S the singular radical. Following D. M. Olson and A. C. Nance [27], we define the special radical class of a ternary semiring as follows: Definition 11. A classM of ternary semirings is called a special radical class ifM is a hereditary class of prime ternary semiring satisfying the following conditions: (1) If S is ternary semi-isomorphic to T and T ∈M , then S ∈M . (2) If I ∈M and I is an ideal of a ternary semiring S, then S/annS(I) ∈M . For prime ternary semirings, we have he following Lemmas. Lemma 6. If the ternary semirings S ≃ T and T is prime, then S is prime. Proof. Suppose that φ is a semi-isomorphism and A, B, C are ideals of S such that ABC = (0). Then φ(A),φ(B),φ(C) are ideals of T and φ(ABC) = (0) ⇒ φ(A)φ(B)φ(C) = (0). Since T is prime, φ(A) = (0) or, φ(B) = (0) or, φ(C) = (0) which implies that A ⊆ Kerφ = (0) ⇒ A = (0) or, B ⊆ Kerφ = (0) ⇒ B = (0), or, C ⊆ Kerφ = (0)⇒ A= (0). Thus, we have either A=(0) or B=(0) or C=(0). Hence, S is prime ternary semiring. In the following lemma, we study the hereditary radical class of ternary semirings. Lemma 7. Let M be a hereditary radical class of prime ternary semirings which satisfies the following properties: “if S ∈M and S is semi-isomorphic to T then T ∈M ”. Then the following conditions are equivalent: (1) If A∈M and A is an ideal of S, then S/annS(A) ∈M ; (2) If A∈M with A an ideal of S and annS(A) = 0, then S ∈M . (3) If A∈M and A is an essential ideal of S, then S ∈M . Proof. The proof follows from Lemma 6. Proposition 9. The class ℘′ of prime nonsingular ternary semirings is a special radical class. Proof. Let S ∈ ℘′ and I be a nonzero ideal of S. Then by Proposition 5, I is a prime non- singular ternary semiring. Since every prime ternary semiring is semiprime, by Proposition 3, Z(I) = I ∩ Z(S) = (0) since Z(S) = 0 as S is nonsingular. Hence, we have I ∈ ℘′. Thus,the class ℘′ is a hereditary class of prime nonsingular ternary semirings. By Theorem 1 and by Lemma 6, the class ℘′ satisfies the property (1) of the definition of special radical class. We now proceed to prove that the class ℘′ is a special radical class. In view of Theorem 1 and Lemma 6, it suffices to prove that if I ∈ ℘′ and I is an essential ideal of a ternary semiring S,then S ∈ ℘′.In order to prove that S is a prime ternary semiring, we let ABC = 0, where A , B and C are three ideals of S. Suppose that A 6= 0, B 6= 0 and C 6= 0. Let A′ = A∩ I ,B′ = B ∩ I T. Dutta, K. Shum, S. Mandal / Eur. J. Pure Appl. Math, 5 (2012), 401-413 411 and C ′ = C ∩ I . Then, A′ , B′ and C ′ are nonzero ideals of I , as I is an essential ideal of S. Now A′B′C ′ ⊆ ABC = 0. Since I is a prime ternary semiring, A′B′C ′ = 0 implies either A′ = 0 or B′ = 0 or C ′ = 0, a contradiction. This shows that S is a prime ternary semiring. Again, Suppose that S is not a nonsingular ternary semiring. Then, by Example 1 and Proposition 2, Z(S) is a nonzero ideal of S and I ∩ Z(S) 6= 0, since I is an essential ideal of S. By Prop 3, Z(I) 6= 0, which is a contradiction. Therefore, S is nonsingular. Thus, S ∈ ℘′. This proves that ℘′ is a special radical class. Finally,we state a theorem of the special radical classes of a ternary ring S. Theorem 9. Let S be a ternary ring. If M is a special radical class of S, then UM is a supernilpotent radical class of S. Proof. In order to show that UM is hereditary, let S ∈ UM and J be a nonzero ideal of S. If J 6∈ UM then there is a nonzero homomorphic image φ(J) of J inM . Let K = kerφ. Then K is a k-ideal of J and we have J/K ≃ φ(J). But φ(J) is prime because it is inM , and by Lemma 6, J/K is prime. Now φ(J) is nonzero, hence J/K 6= (0). Since K is a k-ideal, K is a prime ideal of J and, hence, K is an ideal of S by Lemma 1. Now J/K is a nonzero ideal of S/K , and hence by the property (z), we have (S/K)/ annS(J/K) ∈ M , since by the property (x), J/K ∈ M . If S/K ⊆ annS(J/K), then we have (J/K)3 = (0) which cannot happen because J/K is a prime ternary semiring. Hence, annS(J/K) 6= S/K . Thus, (S/K)/annS(J/K) is a homomorphic image of S which is in M and (S/K)/annS(J/K) 6= (0) as an annihilator ideal is necessarily a k-ideal. However, this is impossible as S ∈ UM . Hence, J ∈ UM and we have shown that UM is a hereditary radical class. Finally, if S is an nilpotent ternary semiring, then φ(S) is nilpotent for any nonzero ho- momorphism φ, and hence φ(S) 6∈ M has no nonzero nilpotent ternary semiring can be prime.Thus, S ∈ UM and hence, UM is supernilpotent. Finally, we state a Theorem for the upper radical class of ternary semirings. Theorem 10. The upper radical class determined by the class ℘′ of prime nonsingular ternary semirings is a supernilpotent radical class. Proof. The proof of the above theorem follows immediately from Proposition 9 and by Theorem 9. Remark 1. We call the upper radical class determined by the class ℘′ the special singular radical class. It is clear that above upper radical class is contained in S . 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