Examples of associative algebras for which the T-space of central polynomials is not finitely based
In 1988 (see [7]), S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A for which 0 ≠ T(A) ⊊ CP(A). V. V. Shchigolev (see [9], 2001) showed that for any field of characteristic zero, every T-space of k0〈X〉 is finitely based, and it follows from this that every T-space of k1〈X〉 is also finitely based. This more than answers Okhitin’s question (in the affirmative) for fields of characteristic zero.For a field of characteristic 2, the infinite-dimensional Grassmann algebras, unitary and nonunitary, are commutative and thus the T-space of central polynomials of each is finitely based.We shall show in the following that if p > 2 and k is an arbitrary field of characteristic p, then neither CP(G0) nor CP(G) is finitely based, thus providing a negative answer to Okhitin’s question.
ON THE GRASSMANN T-SPACE
We study the Grassmann T-space, S3, generated by the commutator [x1,x2,x3] in the free unital associative algebra K 〈x1,x2,… 〉 over a field of characteristic zero. We prove that S3 = S2 ∩ T3, where S2 is the commutator T-space generated by [x1,x2] and T3 is the Grassmann T-ideal generated by S3. We also construct an explicit basis for each vector space S3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x1,…,xn, and deduce the recursive vector space decomposition T3 ∩ Pn = (S3 ∩ Pn) ⊕ (T3 ∩ Pn-1)xn.
Phase diagram and magnetic properties of the diluted fcc system NipMg1-pO.
Starting from Ni and Mg nitrates, about 20 samples of ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O samples (with 0.06\ensuremath{\le}p\ensuremath{\le}0.86) were prepared and x-ray-diffraction studies showed the samples to have the NaCl structure with the lattice constant fitting the expression a(p)=4.2115-0.0340p \AA{}. Temperature-dependent magnetic-susceptibility (\ensuremath{\chi}) studies of the samples were carried out between 1.8 and 600 K using a superconducting-quantum-interference-device magnetometer and N\'eel temperatures ${\mathit{T}}_{\mathit{N}}$'s were determined from the peak in \ensuremath{\partial}(\ensuremath{\chi}T)/\ensuremath{\partial}T. The variation of t=${\mathit{T}}_{\mathit{N}}$(p)/${\mathit{T}}_{\mathit{N}}$(1) vs p is compared with that of ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O. For both systems, the variations for pg0.31 are found to fit the predicted values for a simple-cubic Heisenberg antiferromagnet and a theoretical basis for this anomalous result is advanced. The experimental percolation threshold ${\mathit{p}}_{\mathit{c}}$=0.15\ifmmode\pm\else\textpm\fi{}0.01 and for ${\mathit{p}}_{\mathit{c}}$p0.31, \ensuremath{\chi} below ${\mathit{T}}_{\mathit{N}}$ shows irreversible behavior for the zero-field-cooled and field-cooled cases, suggestive of spin-glass-like behavior also observed in other diluted fcc antiferromagnets, such as ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te. It is suggested that the differences in the t-vs-p variations for p0.31 in ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te may be related to the differences in the ratio of the next-nearest-neighbor to nearest-neighbor exchange constants in these systems.