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.
The finite basis property of T-spaces over fields of characteristic zero
In this paper we prove that any T-space over a field of characteristic zero has a finite basis. This result generalizes Kemer's theorem on the existence of a finite basis for any system of associative identities over a field of characteristic zero.
High-frequency hearing thresholds: effects of age, occupational ultrasound and noise exposure
AbstractPurpose
It has been suggested that high-frequency audiometry (HFA) could represent a useful preventive measure in exposed workers. The aim was to investigate the effects of age, ultrasound and noise on high-frequency hearing thresholds.MethodsWe tested 24 industrial ultrasound-exposed subjects, 113 industrial noise-exposed subjects and 148 non-exposed subjects. Each subject was tested with both conventional-frequency (0.125–8 kHz) and high-frequency (9–18 kHz) audiometry.ResultsThe hearing threshold at high frequency deteriorated as a function of age, especially in subjects more than 30 years old. The ultrasound-exposed subjects had significantly higher hearing thresholds than the non-exposed ones at the high frequencies, being greatest from 10 to 14 kHz. This hearing loss was already significantly evident in subjects with exposure <5 years and increased with years of exposure and advancing age. The noise exposure group had significantly higher hearing thresholds than the non-exposed group at the conventional frequencies 4 and 6 kHz and at the high frequency of 14 kHz. After stratification for age, there was a significant difference between the two groups at 9–10 and 14–15 kHz only for those under 30 years of age.ConclusionMultivariate analysis indicated that age was the primary predictor, and noise and ultrasound exposure the secondary predictors of hearing thresholds in the high-frequency range. The results suggest that HFA could be useful in the early diagnosis of noise-induced hearing loss in younger groups of workers (under 30 years of age).
