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If $S(D)$ $\\vartriangleleft P,$ $\\dim (D)$ can't be controlled. When $R=D[X],$ $I(D)$ $\\vartriangleleft P$ does not imply $I(R)$ $\\vartriangleleft P$ while $I_{t}(D)$ $\\vartriangleleft P$ implies $I_{t}(R)$ $\\vartriangleleft P$ usually. 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If $S(D)$ $\\vartriangleleft P,$ $\\dim (D)$ can't be controlled. When $R=D[X],$ $I(D)$ $\\vartriangleleft P$ does not imply $I(R)$ $\\vartriangleleft P$ while $I_{t}(D)$ $\\vartriangleleft P$ implies $I_{t}(R)$ $\\vartriangleleft P$ usually. 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