The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely ma...The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed,and the order of its unit group is determined. Lastly,another ring isomorphism of M2（Zp） and the quaternion algebra over Zp when p satisfies some particular conditions are presented.展开更多
Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive ...Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive laws does not hold in general. We introduce two ways in which matrix near-rings can be defined and discuss the structure of each. One is as given by Beildeman and the other is as defined by Meldrum. Beildeman defined his matrix near-rings as normal arrays under the operation of matrix multiplication and addition. He showed that we have a matrix near-ring over a near-ring if, and only if, it is a ring. In this case it is not possible to obtain a matrix near-ring from a proper near-ring. Later, in 1986, Meldrum and van der Walt defined matrix near-rings over a near-ring as mappings from the direct sum of n copies of the additive group of the near-ring to itself. In this case it can be shown that a proper near-ring is obtained. We prove several properties, introduce some special matrices and show that a matrix notation can be introduced to make calculations easier, provided that n is small.展开更多
In this paper,we show that every injective Jordan semi-triple multiplicative map on the Hermitian matrices must be surjective,and hence is a Jordan ring isomorphism.
文摘The ring of quaternion over R,denoted by R[i,j,k],is a quaternion algebra. In this paper,the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed,and the order of its unit group is determined. Lastly,another ring isomorphism of M2（Zp） and the quaternion algebra over Zp when p satisfies some particular conditions are presented.
文摘Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive laws does not hold in general. We introduce two ways in which matrix near-rings can be defined and discuss the structure of each. One is as given by Beildeman and the other is as defined by Meldrum. Beildeman defined his matrix near-rings as normal arrays under the operation of matrix multiplication and addition. He showed that we have a matrix near-ring over a near-ring if, and only if, it is a ring. In this case it is not possible to obtain a matrix near-ring from a proper near-ring. Later, in 1986, Meldrum and van der Walt defined matrix near-rings over a near-ring as mappings from the direct sum of n copies of the additive group of the near-ring to itself. In this case it can be shown that a proper near-ring is obtained. We prove several properties, introduce some special matrices and show that a matrix notation can be introduced to make calculations easier, provided that n is small.
基金Supported by the National Natural Science Foundation of China (Grant Nos.11001194 10771157)the Natural Science Foundation of Shanxi Province (Grant No.2009021002)
文摘In this paper,we show that every injective Jordan semi-triple multiplicative map on the Hermitian matrices must be surjective,and hence is a Jordan ring isomorphism.
