the free associative algebra  k < a_ij , b_ji, i = 1,2,...,n , j = 1,2,...,m >  n < m . over the field k . In this free ring 

we consider the two sided ideal I generated by the following elements: 

 

The residual ring V_n,m = k <  a_ij , b_ji, i = 1,2,...,n.j = 1,2,...,m > I  is a non-IBN ring.  The construction 

is due to P.M.Cohn.Therefore for each pair of positive integers (n,m)  with n<m , there corresponds 

a non-IBN ring. 

                       In this work we have studied the structure of the V_n,m rings. In Chapter I  we have shown 

that the rings V_1,m do not have any proper two sided ideal other than the zero ideal itself. In Chapter II we 

have constructed a complete set of representatives of  the V_n,m rings in the free associative algebra , 

by which we have studied the additive structure and the multiplicative structure of  the V_n,m rings. 

In Chapter III  we have proved that the V_n,m rings for n # 1, do not have any zero divisors. This has 

been shown by  proving that if the two sided ideal I of the free associative algebra contains a product 

xy, then either  or . The fact that the rings V_n,m having no divisors of zeros  proves that 

the rings V_1,m cannot be imbedded isomorphically into any of the V_n,m rings for n # 1. Finally we have 

shown that the rings V_1,1+2V_(2r-1) can be imbedded isomorphically into the rings V_1,2r for v,r = 1,2,... 

Moreover the rings V_n,n+2v_(2r-1) can  homomorphically be imbedded into the rings V_n,n+(2r-1) for v,r = 1,2,.... 

Where we have  shown these facts by proving that for any pair of positive integers  (n,m) the rings V_n,2m-n 

can homomorphically imbedded into rings V_n,m .