This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over SLn(Z(p)). Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over SL3(Z(p)), Forum Math. 31 (2019), no. 1,111-147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SLn(Z(p)), Internat. J. Algebra Comput. 20 (2010), no. 8, 1021-1039] simultaneously. Let n (n >= 2) be a positive integer. Let p (p > 2) be a prime integer, Z(p) the ring of p-adic integers and F-p the finite filed of p elements. Let G =Gamma(1)(SLn(Z(p))) be the first congruence subgroup of the special linear group SLn(Z(p)) and Omega(G) the mod-p Iwasawa algebra of G defined over F-p. By a purely computational approach, for each nonzero element W is an element of Omega(G), we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under "nice prime" condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 (2008), no. 1, 259-275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 (2008), no. 3, 865-901]. This paper currently provides a new proof without the "nice prime" condition. As a consequence of the above-mentioned main result, we observe that the center of Omega(G) is trivial.
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