QUOTIENT SEMINEAR-RINGS OF THE ENDOMORPHISM OF SEMINEAR-RINGS

A seminear-ring is a generalization of ring. In ring theory, if  is a ring with the multiplicative identity, then the endomorphism module  is isomorphic to . Let  be a seminear-ring. Here, we can construct the set of endomorphism from  to itself denoted by . We show that if  is a seminear-ring, then is also a seminear-ring over addition and composition function. We will apply the congruence relation to get the quotient seminear-ring endomorphism. Furthermore, we show the relation between c-ideal and congruence relations. So, we can construct the quotient seminear-ring endomorphism with a c-ideal

The Ideal Over Semiring of the Non-Negative Integer

Assumed that (S,+,.) is a semiring. Semiring is a algebra structure as a generalization of a ring. A set I⊆S is called an ideal over semiring S if for any α,β∈I, we have α-β∈I and sα=αs∈I for every s in semiring S.  Based on this definition, there is a special condition namely prime ideal P, when for any αβ∈P, then we could prove that α or β are elements of ideal P. Furthermore, an ideal I of S is irreducible if Ia is an intersection ideal from any ideal A and B on S, then I=A or I=B. We also know the strongly notion of the irreducible concept. The ideal I of S is a strongly irreducible ideal when I is a subset of the intersection of A and B (ideal of S), then I is a subset of A, or I is a subset of B. In this paper, we discussed the characteristics of the semiring of the non-negative integer set. We showed that pZ^+ is an ideal of semiring of the non-negative integer Z^+ over addition and multiplication. We find a characteristic that 〖pZ〗^+  is a prime ideal and also a strongly irreducible ideal of the semiring Z^+ with p is a prime number

IDEAL TAK TEREDUKSI KUAT ATAS SEMIRING KOMUTATIF

Abstrak. Himpunan tak kosong yang dilengkapi suatu operasi biner yang bersifat asosiatif disebut semigrup. Setiap semigrup yang memuat elemen identitas didalamnya disebut monoid. Selanjutnya, grup adalah sebuah monoid dimana setiap elemennya mempunyai elemen invers. Setiap grup yang memenuhi sifat komutatif disebut grup komutatif. Ring (R,+,.) didefinisikan sebagai himpunan tak kosong yang dilengkapi dengan dua operasi biner yaitu penjumlahan dan pergandaan serta memenuhi beberapa aksioma tertentu diantaranya (R,+) adalah grup komutatif, (R,.) semigrup dan (R,+,.) memenuhi hukum distributif kiri beserta distributif kanan Struktur aljabar semiring merupakan generalisasi dari ring dengan mengurangi keberadaanelemen invers pada operasi penjumlahan. Semiring disebut semiring komutatif asalkanoperasi pergandaan pada semiring bersifat komutatif. Ideal pada semiring didefinisikan dengan cara yang sejalan dengan ideal pada ring. Suatu ideal pada sebuah semiring dikatakan tak tereduksi jika ideal adalah hasil irisan antara ideal A dan B maka I=A atau I=B dan suatu ideal pada sebuah semiring dikatakan tak tereduksi kuat jika ideal adalah himpunan bagian dari hasil irisan antara ideal A dan B maka I=A atau I=B. Pada paper ini diperoleh hasil, setiap ideal tak tereduksi kuat merupakan ideal tak tereduksi.&nbsp