On the Complexity of Radicals in Noncommutative Rings
نویسنده
چکیده
This article expands upon the recent work by Downey, Lempp, and Mileti [3], who classified the complexity of the nilradical and Jacobson radical of commutative rings in terms of the arithmetical hierarchy. Let R be a computable (not necessarily commutative) ring with identity. Then it follows from the definitions that the prime radical of R is Π1, and the Levitzki radical of R is Π 0 2. We show that these upper bounds for the complexity of the prime and Levitzki radicals are optimal by constructing two noncommutative computable rings with identity, such that the prime radical of one is Π1-complete, while the Levitzki radical of the other is Π 0 2-complete.
منابع مشابه
On zero-divisor graphs of quotient rings and complemented zero-divisor graphs
For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...
متن کامل*-σ-biderivations on *-rings
Bresar in 1993 proved that each biderivation on a noncommutative prime ring is a multiple of a commutatot. A result of it is a characterization of commuting additive mappings, because each commuting additive map give rise to a biderivation. Then in 1995, he investigated biderivations, generalized biderivations and sigma-biderivations on a prime ring and generalized the results of derivations fo...
متن کاملON ANNIHILATOR PROPERTIES OF INVERSE SKEW POWER SERIES RINGS
Let $alpha$ be an automorphism of a ring $R$. The authors [On skewinverse Laurent-serieswise Armendariz rings, Comm. Algebra 40(1)(2012) 138-156] applied the concept of Armendariz rings to inverseskew Laurent series rings and introduced skew inverseLaurent-serieswise Armendariz rings. In this article, we study on aspecial type of these rings and introduce strongly Armendariz ringsof inverse ske...
متن کاملOn Generalized Reed-Solomon Codes Over Commutative and Noncommutative Rings
In this paper we study generalized Reed-Solomon codes (GRS codes) over commutative, noncommutative rings, show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context and we investigate their complexities. Under some hypothesis, the study of noncommutative generalized Reed-Solomon codes over finite rings leads to the fact that GRS code over commutat...
متن کاملSinglet scalar dark matter in noncommutative space
In this paper, we examine the singlet scalar dark matter annihilation to becoming the Standard Model particles in the non-commutative space. In the recent decades, many candidates of dark matter have been offered, but our information about the nature of dark matter is still limited. There are such particle candidates as scalar matetr, fermion, boson, gauge boson, etc.; however, they have nei...
متن کامل