Isomorphisms of unitary groups over associative rings

Automorphisms of unitary linear groups u (n,,q) over the ring are found to be standard on an elementary unitary subgroup in the case when the hyperbolic rank of the form q is strictly greater than one and n5 (for a commutative ring, n4).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR, Vol. 132, pp. 97–109, 1983.     

Finite presentability of Steinberg groups over group rings

LetA be a finitely generated commutative -algebra with Krull dimensiond, and let be an arbitrary finite group. It is proved that the Steinberg groupSt _n (A) is finitely presented whenevern4. If, in addition,nd+3, andK ₁ (A) andK ₂ (A) are finitely generated, thenE _n (A) andGL _n (A) are finitely presented.The Project supported by National Natural Science Foundation of China.     

Homomorphisms of linear groups over commutative rings

Translated from Matematicheskie Zametki, Vol. 46, No. 5, pp. 50–61, November, 1989.     

Representations of quantum groups defined over commutative rings

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules. The notion of an induced invariant form is introduced and a setting in described where all invariant forms are induced     

Representations of quantum groups defined over commutative rings II

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.     

The structure of symplectic groups over arbitrary commutative rings

LetR be an arbitrary commutative ring, andn be an integer ≥3. It is proved for any idealJ ofR thatESp _2n(R,J)=[ESp _2n(R),ESp _2n(J)]=[ESp _2n(R),ESp _2n(R,J)] =[ESp _2n(R),GSp _2n(R,J)]=[Sp _2n(R),ESp _2n(R,J)]. Furthermore, the problem of normal subgroups ofSp _2n(R) has an affirmative solution if and only ifaR=a ²R+2aR for eacha inR. This covers the relevant results of [4], [8], [10], [12] and [13]. Project Supported by the Science Fund of the Chinese Academy of Sciences     

Quaternion algebras over commutative rings

Translated from Matematicheskie Zametki, Vol. 53, No. 2, pp. 126–131, February, 1993.     

Polynomial rings over Goldie-Kerr commutative rings II

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).

     

Linear equations over commutative rings

A generalized rank (McCoy rank) of a matrix with entries in a commutative ring R with identity is discussed. Some necessary and sufficient conditions for the solvability of the linear equation Ax = b are derived, where x, b are vectors and A is a matrix with entries in either a Noetherian full quotient ring or a zero dimensional ring.