Examples of noncatenary rings

MATH 436 Notes: Examples of Rings

i∈N R = {(a0, a1, a2, . . . )|aj ∈ R} is a countable direct product of (R,+) with itself. Thus the elements of R[[x]] are arbitrary sequences with entries in R where addition is componentwise. We define a multiplication ⋆ via (a0, a1, . . . ) ⋆ (b0, b1, . . . ) = (c0, c1, . . . ) where cn = ∑n k=0 akbn−k for all n ∈ N. It is easy to check that the sequence (1, 0, 0, . . . ) is a two-sided ident...

Functionally recursive rings of matrices-Two examples

We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 ...

Three Examples of Gröbner Reduction over Noncommutative Rings

On Some Examples of Fullerenes with Heptagonal Rings

ABSTRACT Several works on nonclassical fullerenes with heptagons have mainly considered the case with just one heptagon. In this context the isolated pentagon rule is not satisfied. The study of nonclassical fullerenes is important because some of them are more stable than the corresponding classical isomers with the same number of pentagonal bonds. We present several nonclassical fullerenes wi...

Differential Polynomial Rings of Triangular Matrix Rings