Summary: A soluble group \(G\) is said to be rigid if it contains a normal series of the form \(G=G_1>G_2>\cdots>G_p>G_{ p+1}=1\), whose quotients \(G_i/G_{i+1}\) are Abelian and are torsion-free when treated as right \(\mathbb Z[G/G_i]\)-modules. Free soluble groups are important examples of rigid groups. A rigid group \(G\) is divisible if elements of a quotient \(G_i/G_{i+1}\) are divisible by nonzero elements of the ring \(\mathbb Z[G/G_i]\), or, in other words, \(G_i/G_{i+1}\) is a vector space over the division ring \(Q(G/G_i)\) of quotients of that ring. A rigid group \(G\) is decomposed if it splits into a semidirect product \(A_1A_2\cdots A_p\) of Abelian groups \(A_i\cong G_i/G_{i+1}\). A decomposed divisible rigid group is uniquely defined by cardinalities \(\alpha_i\) of bases of suitable vector spaces \(A_i\), and we denote it by \(M(\alpha_1,\dots,\alpha_p)\). The concept of a rigid group appeared in [A. Myasnikov and the author, J. Algebra 324, No. 10, 2814-2831 (2010; Zbl 1234.20052)], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [Algebra Logic 48, No. 2, 147-160 (2009); translation from Algebra Logika 48, No. 2, 258-279 (2009; Zbl 1245.20036)], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group \(M(\alpha_1,\dots,\alpha_p)\). Our present goal is to derive important information directly about algebraic geometry over \(M(\alpha_1,\dots,\alpha_p)\). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over \(M(\alpha_1,\dots,\alpha_p)\) using the language of equations.