Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.
Enantioselective synthesis of cyclobutane derivatives is still a challenging topic in asymmetric synthesis. [2+2]-Cycloaddition and skeleton rearrangement are two primary strategies to this end. Recently, functionalization of cyclobutanones and cyclobutenones, which are readily available via [2+2]-cycloadditions as prochiral substrates, has emerged as a powerful tool to access versatile four-membered ring compounds. Herein, we summarize some recent advances in these areas from our and other groups.
Ultraviolet irradiation of 3-(p-anisyl)-2-cyclohexenone and 3-(p-nitrophenyl)-2-cyclohexenone has been found to give a single photodimer in each case, the structures of which have been established as the cis–anti–cis head-to-head cyclobutane derivatives XI and XIII, respectively.
For a ring [Formula: see text], an extension ring [Formula: see text], and a fixed right [Formula: see text]-module [Formula: see text], we prove the induced left [Formula: see text]-module [Formula: see text] is a Wakamatsu tilting module when [Formula: see text] is a Wakamatsu tilting module.
Normal Bases on Galois Ring Extensions
