We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse). An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .
<span>In this work, new discrete wavelets were derived Hermite polynomials for obtained discrete hermite wavelet transformation (DHWT), and their efficiency for use in image processing is demonstrated by proving the realization of important theorems. Moreover, the role of the new and proposed waveforms in their effective effect in placing the watermark with the color image is clarified, and a program was created using MATLAB software by creating a subprogram for constructing the new wavelet and proving its efficiency with an analytic image. The process is repeated using DHWT to analyze the image. The color image has been subjected to various attacks after which the watermark is retrieved from the image after comparing it with the proposed algorithm and it has proven its power faster and better than the previously suggested methods. The final conclusion shows that using new wavelets DHWT better peak signal of noise ratio (PSNR)s can be obtained and that the proposed algorithm fills in better the lack of awareness of the watermark and its strength under different attacks.</span>
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