2-absorbing primary ideals of amalgamations

One of the most important problems of commutative algebra is studying over new ideal classes and investigating their in special rings. We can give prime ideals as an example. Prime ideals are very important in commutative rings. Therefore, generalizations of the definition of prime ideals are quite common. For example, if I is a nonzero ideal of the ring R and for elements a,b,c ∈R when abc∈I such that ab∈I or bc∈I or ac∈I, then the ideal I is called a 2-absorbing ideal of R. Another example is if I is an ideal of the ring R and for a_1,a_2,a_3,…,a_(n+1)∈R. If the product of n elements of a_i is contained in the ideal I when a_1,a_2,a_3,…,a_(n+1)∈I, then the ideal I is called an n-absorbing ideal. Similarly, if I is an ideal of the ring R and for elements a,b,c ∈R, when abc∈I such that ab∈I or ac∈√I or bc∈√I is, then the ideal I is called a 2-absorbing primary ideal of the ring R. These ideals have been studied in different rings, such as local rings, Noetherian rings, Artinian rings or trivial extensions. Similarly, some of these ideals have been studied in different articles for the Amalgamated ring which is a new ring. In this study, we will try to determine the 2-absorbing primary ideals in the Amalgamated ring. Let A and B be two rings, let J be an ideal of B and I be an ideal of A, and let f:A→B be a ring homomorphism. A ⋈^f J = {(a,f(a) + j): a ∈ A,j ∈ J} for all a,a_1, a_2∈A and j,j_1,j_2∈J (a_1,f(a_1 )+j_1 )+(a_2,f(a_2 )+j_2 )≔(a_1 〖+a〗_2,f(a_1 )+〖f(a〗_2 )+j_1+j_2) and (a_1,f(a_1 )+j_1 )(a_2,f(a_2 )+j_2 )≔(a_1 a_2,f(a_1 a_2 )+〖f(a〗_2 ) j_1+f(a_1)j_2+j_1 j_2) is a ring under operations. This ring is a subring of the ring A×B. This ring is called the amalgamation of A and B under f along the ideal J. Also, the set I⋈^ƒ J∶={ (i,f(i)+j) ∶i∈I,j∈J} is an ideal of the ring A⋈^ƒ J. Our main motivation in this thesis is to show when the ideal I⋈^ƒ J is a 2-absorbing primary ideal in the ring A⋈^ƒ J under the given operations. First, we give the prime ideals of this ring, the radical of the ideal and the relation of the prime ideals to the 2-absorbing primary ideals. We obtain some theorems, examples and results related to this. In particular, we give results on the 2-absorbing primary ideals of our ring when it is a local ring, a divided ring and a chain ring.