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The paper aims to present a new primary ideal in a commutative ring R with nonzero identity element. We introduce the new primary ideal as an n-1-absorbing primary ideal that is a generalization of both primary and 1-absorbing primary ideals. We propose to achieve two goals with this paper. Firstly, we study and characterize some essential properties of an n-1-absorbing primary ideals and figure out the relations between the other types of ideals such as prime, 1- absorbing primary and irreducible ideals. Then, we classify some special rings that admit an n-1-absorbing primary ideal. We provide the results by introducing some examples. [ABSTRACT FROM AUTHOR]

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1x p n−1/xn(any q n−2 +bnynyn−1),/ yn+1 = xnxn−1y q n−1 yn(cnx p n−2 +dnxnxn−1), p,q ∈ N,n ∈ N0 where the initial values x−i, y−i,i = 0,1,2 and the parameters (an)n∈N0,(bn)n∈N0,(cn)n∈N0, (dn)n∈N0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented. [ABSTRACT FROM AUTHOR]