Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings
Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings
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右侧理想和多重线性多项式在素环上的推导
DOI：
10.4171/rsmup/121-15
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Rajneesh Sharma
中科院分区：
文献类型：
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作者：
B. Dhara;Rajneesh Sharma
设R是特征为R6 = 2的结合素环，中心Z（R）和扩张质心C，f（x1，. . .，xn）是C上n个非交换变元的非零多线性多项式，d是R的非零导子，ρ是R的非零右理想.我们证明了：（i）如果[d（f（x1，. . .，xn）），f（x1，. . .，xn）] = 0，对于所有x1，. . .，xn ∈ ρ，则对RC的基柱中的幂等元e，ρC = eRC，f（x1，. . .，xn）在eRCe中是中心值的，除非d是由B ∈ Q导出的内导子使得B = 0且Bρ = 0;（ii）如果[d（f（x1，. . .，xn）），f（x1，. . .，xn）] ∈ Z（R），对所有的x1，. . .，xn ∈ ρ，则对RC的基柱中的幂等元e，且f（x1，. . .，xn）在eRCe中是中心的，或者eRCe满足标准恒等式S4（x1，x2，x3，x4），除非d是由B ∈ Q导出的内导子，使得B = 0并且Bρ = 0。数学学科分类：16 W25、16 R50、16 N60。
Let R be an associative prime ring of char R 6= 2 with center Z(R) and extended centroid C, f(x1, . . . , xn) a nonzero multilinear polynomial over C in n noncommuting variables, d a nonzero derivation of R and ρ a nonzero right ideal of R. We prove that: (i) if [d(f(x1, . . . , xn)), f(x1, . . . , xn)] = 0 for all x1, . . . , xn ∈ ρ then ρC = eRC for some idempotent element e in the socle of RC and f(x1, . . . , xn) is central-valued in eRCe unless d is an inner derivation induced by b ∈ Q such that b = 0 and bρ = 0; (ii) if [d(f(x1, . . . , xn)), f(x1, . . . , xn)] ∈ Z(R) for all x1, . . . , xn ∈ ρ then ρC = eRC for some idempotent element e in the socle of RC and either f(x1, . . . , xn) is central in eRCe or eRCe satisfies the standard identity S4(x1, x2, x3, x4) unless d is an inner derivation induced by b ∈ Q such that b = 0 and bρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60.