In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A co...In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.展开更多
In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Fi...In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.展开更多
Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference 'Wang Shianghaw, On quasi-valuation ring, Northeast People's Univ. Natur...Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference 'Wang Shianghaw, On quasi-valuation ring, Northeast People's Univ. Natur. Sci. J., (1)(1957), 27-40', when the quotient field of R is an algebraic number field or an algebraic function field, and we obtain a characterization of quasi-valuation rings.展开更多
Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b i...Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.展开更多
文摘In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.
文摘In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.
文摘Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference 'Wang Shianghaw, On quasi-valuation ring, Northeast People's Univ. Natur. Sci. J., (1)(1957), 27-40', when the quotient field of R is an algebraic number field or an algebraic function field, and we obtain a characterization of quasi-valuation rings.
文摘Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.
