For a ring R(not necessarily commutative)with identity,the comaximal graph of R,denoted byΩ(R),is a graph whose vertices are all the nonunit elements of R,and two distinct vertices a and b are adjacent if and only if...For a ring R(not necessarily commutative)with identity,the comaximal graph of R,denoted byΩ(R),is a graph whose vertices are all the nonunit elements of R,and two distinct vertices a and b are adjacent if and only if Ra+Rb=R.In this paper we consider a subgraphΩ_(1)(R)ofΩ(R)induced by R\Uℓ(R),where Uℓ(R)is the set of all left-invertible elements of R.We characterize those rings R for whichΩ_(1)(R)\J(R)is a complete graph or a star graph,where J(R)is the Jacobson radical of R.We investigate the clique number and the chromatic number of the graphΩ_(1)(R)\J(R),and we prove that if every left ideal of R is symmetric,then this graph is connected and its diameter is at most 3.Moreover,we completely characterize the diameter ofΩ_(1)(R)\J(R).We also investigate the properties of R whenΩ_(1)(R)is a split graph.展开更多
Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a sy...Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a system of partial differential equations.展开更多
In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R ...In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.展开更多
基金This research was supported by NSFC(12071484,11871479)Hunan Provincial Natural Science Foundation(2020JJ4675,2018JJ2479)the Research Fund of Beijing Information Science and Technology University(2025030).
文摘For a ring R(not necessarily commutative)with identity,the comaximal graph of R,denoted byΩ(R),is a graph whose vertices are all the nonunit elements of R,and two distinct vertices a and b are adjacent if and only if Ra+Rb=R.In this paper we consider a subgraphΩ_(1)(R)ofΩ(R)induced by R\Uℓ(R),where Uℓ(R)is the set of all left-invertible elements of R.We characterize those rings R for whichΩ_(1)(R)\J(R)is a complete graph or a star graph,where J(R)is the Jacobson radical of R.We investigate the clique number and the chromatic number of the graphΩ_(1)(R)\J(R),and we prove that if every left ideal of R is symmetric,then this graph is connected and its diameter is at most 3.Moreover,we completely characterize the diameter ofΩ_(1)(R)\J(R).We also investigate the properties of R whenΩ_(1)(R)is a split graph.
文摘Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a system of partial differential equations.
基金supported by National Natural Science Foundation of China(Grant No.12301001)。
文摘In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.
