Coherence of Polynomial Rings over Semisimple Algebraic Algebras

Triangulaizability of Algebras Over Division Rings

On Algebraic Shift Equivalence of Matrices over Polynomial Rings

The paper studies algebraic shift equivalence of matrices over n-variable polynomial rings over a principal ideal domain D(n ≤ 2). It is proved that in the case n = 1, every non-nilpotent matrix over D[x] is algebraically strong shift equivalent to a nonsingular matrix. In the case n = 2, an example of non-nilpotent matrix over R[x, y, z] = R[x][y, z], which can not be algebraically shift equiv...

Gray Images of Constacyclic Codes over some Polynomial Residue Rings

Let  be the quotient ring    where  is the finite field of size   and  is a positive integer. A Gray map  of length  over  is a special map from  to ( . The Gray map   is said to be a ( )-Gray map if the image of any -constacyclic code over    is a -constacyclic code over the field   . In this paper we investigate the existence of   ( )-Gray maps over . In this direction, we find an equivalent ...

Polynomial Rings over Pseudovaluation Rings

Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x ∈ P for any P ∈ Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring. (2) Let R be a σ-divided ring such that x ∈ P for any P ∈ Spec(R[x,σ]). Then R[x,σ] is also a σ-divided ring. Let now R be a commutative Noetherian Q-algebra (Q i...

Lecture 5: Semisimple Lie Algebras over C

In this lecture I will explain the classification of finite dimensional semisimple Lie algebras over C. Semisimple Lie algebras are defined similarly to semisimple finite dimensional associative algebras but are far more interesting and rich. The classification reduces to that of simple Lie algebras (i.e., Lie algebras with non-zero bracket and no proper ideals). The classification (initially d...