Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) https://internationalpubls.com 1180 A Note on Radicals of Semiring of Matrices Dr. Manohar B. Bhagirath1, Dr. Narendrakumar R. Dasre2, Dr. Pritam Gujarathi-Wani3* 1Associate Professor, Head, Department of Mathematics, Annasaheb Vartak College of Arts, Science and Commerce, Vasai, Dist. Palghar - 401202. Email-id : manoharbhagirath@gmail.com 2Associate Professor, Head, Department of Engineering Sciences, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai-400706, Email-id : narendasre@rait.ac.in 3*Assistant Professor, Department of Engineering Sciences, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai-400706, Email-id : pritam.wani@rait.ac.in *Corresponding Author: Dr. Pritam Gujarathi-Wani Email-id : pritam.wani@rait.ac.in Article History: Received:01/11/2024 Revised: 06/12/2024 Accepted: 30/12/2024 Abstract: In this article we introduce and investigate radicals of semiring of matrices. We establish that if R is a radical class which is (right or left)-hereditary and (right or left)-strong, then R has the property that the R-radical of the semiring of matrices of order n over a semiring R is equal to the semiring of n×n-matrices over the semiring R(R). Keywords: Semirings, Radical classes, Semiring of Matrices. 1. Introduction Throughout this article semirings will be associative, not necessariliy with unity element and radicals in the sense of Kurosh Amitusar as defined in [9]. In this article we have introduced and investigated radicals of semiring of matrices and of polynomial semirings. In this article, we have shown that if R is a radical class which is (right or left)- hereditary and (right or left)-strong, then R has the property that the R-radical of the semiring of matrices of order n over a semiring R is equal to the semiring of matrices of order n over the semiring R(R). The interrelation and independence of polynomial extensibility of radical and semisimple classes and of the Amitsur property are investigated for associative rings in [12]. In this article, we have tried to generalize some results for semirings. Throughout this article ↦ stands for onto homomorphism and ⊲ stands for an ideal of a semiring R. For details of semiring theory and more on radical theory for associative semirings the readers are referred to [1], [2], [6], [8] and [10]. 2. Radical of Semiring of Matrices. Definition 2.1. The additive semigroup (R, +) of a semiring R will be denoted by R + and for an abelian semigroup R +, we may define always a semiring R 0 with zero multiplication, called a zero- semiring by the rule xy = 0 for all x, y ∈ R. Definition 2.2. A radical R is called an A-radical if for any semiring R ∈ R and any additive homomorphism f : R → S such that f (R) is a subsemiring of S also f (S) ∈ R. Definition 2.3. A semiring R is said to be simple if it has no proper ideals. Definition 2.4. If M is the class of all simple semirings with unity, then UM is called the Brown McCoy radical class. Proposition 2.5. Let ρ be a regular class of semirings. The upper radical Uρ is hereditary if and only if ρ satisfies the condition (I) : if 0 ≠ 𝐼 ⊲ R and there is a I ↦ T such that 0 ≠ T ∈ ρ, then there exists an R ↦ S such that 0 ≠ S ∈ ρ. mailto:manoharbhagirath@gmail.com mailto:%20narendasre@rait.ac.in mailto:pritam.wani@rait.ac.in mailto:pritam.wani@rait.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) https://internationalpubls.com 1181 Proof. Suppose that Uρ is hereditary. If 0 ≠ 𝐼 ⊲ R and there is a I ↦ 𝑇 such that 0 ≠ T ∈ ρ, then there exists an R ↦ S such that 0 ≠ S ∈ ρ. Conversely, by (I), I ∉ Uρ ⇒ R ∉ Uρ. But then R ∈ Uρ ⇒ I ∈ Uρ, hence Uρ is hereditary. Proposition 2.6. If K is a subtractive ideal in an ideal I of a semiring R and I/K is a semiring with unity element, then K is an ideal in R. Proof. Let e + K be the identity in I/K. Then any a ∈ R and any k ∈ K. Then we have ak ∈ I as K ⊲ I. Therefore ak + K = (e + K)(ak + K) = e(ak) + K = (ea)k + K = K. ak + K = K ⇒ ak + k1 = k2 ∈ K for some k1, k2 ∈ K ⇒ ak ∈ K (K is subtractive). Similarly we get ka ∈ K and K ⊲ R. Proposition 2.7. If ρ is any regular class of semirings with unity element, then the upper radical Uρ is hereditary. Proof. We must show that ρ satisfies condition (I) of Proposition 2.5 that I is an ideal in a semiring R such that I has a non-zero homomorphic image C = I/K,(K is subtractive) in ρ. Then I/K has a unity and by Proposition 2.6, K is an ideal of R. Since I/K has a unity, it must be a direct summed of R/K. Then R/K can be mapped homomorphically to I/K and this is in ρ and non-zero. Hence (I) holds and Uρ is hereditary. Theorem 2.8. Brown McCoy radical is hereditary. Theorem 2.9. Dorroh’s Extension Theorem : Every semiring R can be embedded as an ideal into a semiring R with unity element. Proof. On the set R’ = { (a, n) / a ∈ R and n ∈ Z+ ∪ { 0}}. Define (a, n) + (a’, n’ ) = (a + a’, n + n’ ) and (a, n)( a’, n’) = (aa’ + n’a +nb’, nn’ ). R’ is a semiring with unity (0, 1) and R ≅ (R, 0) ⊲ R’ Note : The ring R’ is refereed to be the Dorroh’s extension. We shall use the following notations. If R is a semiring and n is a positive integer. R (n) denotes the semiring of matrices of order n over R. For i, j ∈ {1, 2, 3, ..., n}, R ( j ) denotes the subsemiring of R (n) consisting of all matrices with elements from R in the (i, j)th position and with 0’s elsewhere. For i ∈ {1, 2, 3, ..., n}, we define R (i) as the right ideal ∑ R(𝑖𝑗)𝑛 𝑗=1 of R (n) and we define Li as the left ideal ∑ R(𝑘𝑖)𝑛 𝑘=1 of R (n). If x ∈ R and J is a non-empty subset of {1, 2, 3, ..., n} with i ∈ J, then Xj (i)(x) denotes the n × n matrix with x in the (i, j)-th position for all j ∈ J and with 0’s elsewhere. Then X(i) = ⋃ 𝑥∈𝑅 Xj (i)(x) is a left ideal of the semiring R (n). Moreover R ≅XJ (i, x) under the natural map. If R is the Brown McCoy radical, then R is hereditary and satisfies R(R (n)) = (R(R))(n). Proposition 2.10. If I ⊲ R, then I (n). Moreover if I is a subtractive ideal, then so is I (n). Proposition 2.11. If R is a semiring with unity element and K ⊲ R (n), then K = I (n) with some I ⊲ R. Proposition 2.12. If R is a radical, then R(R(n)) = I (n) for some ideal I of R and for every semiring R. Proof. For any radical R and for any semiring R, if I ⊲ R, then R(I) ⊲ R. If I ⊲ R, then I (n) ⊲ R (n) ⇒ R(I (n)) ⊲ R(n) ⇒ R(I (n)) ⊆ R(R (n)). In particular, R(R(n)) is an ideal in R(n). Therefore R(R(n)) = I (n) for some ideal I in R’, where R’ is Dorroh’s extension. But I (n) ⊆ R (n) ⇒ I ⊆ R ⇒ I ⊲ R. Theorem 2.13. Let R be a radical class, let R be a semiring, and let n be a positive integer. The following statements are equivalent. (1) If R ∈ R, then R (n) ∈ R. (2) (R(R))(n) ⊆ R(R (n)). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) https://internationalpubls.com 1182 Proof. Now R(R) ∈ R so that by (1) (R(R))(n) ∈ R. Since (R(R))(n) is an ideal in R (n), hence (R(R))(n) ⊆ R(R (n)). Now R∈ R implies that R(R) = R. So that R (n) = (R(R))(n). By (2) (R(R))(n) ⊆ R(R (n)). Hence R(n)= R(Rn) and Rn ∈ R. Theorem 2.14. Let R be a radical class, let R be a semiring, and let n be a positive integer. The following statements are equivalent. 1. If R (n) ∈ R, then R ∈ R. 2. R(R (n)) ⊆ (R(R))(n) Proof. By Proposition 2.12 R(R (n)) = I(n), for some ideal I of R. From (1) we have I ∈ R. Hence I ⊆ R(R) and so R(R (n)) = I (n) ⊆ (R(R))(n). Now R (n) ∈ R implies that R(R (n)) = R (n). Thus by (2), R (n) = R(R (n)) ⊆ (R(R))(n) and so R (n) = (R(R))(n). Hence R(R) = R and R ∈ R. Theorem 2.15. Let R be a strong radical class. Then R ∈ R implies R (n) ∈ R. Proof. Let n > 1, let i ∈ {1, 2, 3, ..., n} be fixed and let j ∈ {1, 2, 3, ..., n} with i≠ j. Set J = {i, j}. Then Xj (i)(∈ R. Since R is strong, therefore XJ (i) ⊆ R(R (i)). Setting K = {i}. We like wise obtain XK (i) ⊆ R(R(i)). Hence XJ (i) + XK (i) ⊆ R(R(i)). Since i ≠ j. and j was otherwise arbitary, then Ri ⊆ R(R (i)). Now R (i)is R-right ideal of R(n) so that, since R is strong, we have R(i) ⊆ R(R (n)). This is true for j ∈ {1, 2, 3, ..., n}. We obtain ∑ R(𝑖)𝑛 𝑖=1 ⊆ R(R(n)). Hence R (n) = R(R (n)) and R (n) ∈ R. Theorem2.16. Let R be a hereditary (or right hereditary) radical class. Then R (n) ∈ R implies R ∈ R. Theorem2.17. Let R be a hereditary and left-strong (right strong) radical class. Then R ∈ R implies R+ ∈ R. Theorem 2.18. Let R be a hereditary and left-strong (right strong) radical class. Then R ∈ R implies R (n)∈ R. Theorem 2.19. Let R be a radical class which is (left or right)-hereditary and (left or right) -strong. Then R(R (n)) = (R(R (n)))n. Acknowledgement: I would like to extend my heartfelt gratitude to Dr. Rajendra P. Deore, our respected mentor, for his invaluable guidance and unwavering support throughout this research work. References [1] Al-Thani H. M. J. Weak Radical Classes, Tamkang Journal of Mathematics, 35(4), 359-369 (2004). [2] Dutta T. K. and Das M. L. Normal Radical Class of Semirings, Siutheast Asian Bulletin of Mathematics, 35, 389-400 (2011). [3] B. J. Gardner, A note on radicals and polynomial rings, Math. Scand., 31 (1972), 83-88. [4] Gardner B. J. and Wiegandt R. Radical Theory of Rings, Marcel Dekker, (2004). [5] B. J. Gardner, Radicals of Abelian Groups and Associative Rings , Acta Math. Acad. Sci. Hung., 24(3-4), 259-268 (1973). [6] Golan J. S. Semirings and their Applications, Kluwer Academic Publisher (1999). [7] N. V. Loi, R. 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