Let \(A=A_1\oplus\cdots\oplus A_r\) be a decomposition of the associative algebra \(A\) as a direct sum of its vector subspaces \(A_i\). This decomposition is regular if for any choice of the indices \(i_j\) one has \(A_{i_1}\cdots A_{i_n}\neq 0\), and furthermore for every \(i\) and \(j\) and every \(x_i\in A_i\), \(x_j\in A_j\) one has \(x_ix_j=\theta_{ij}x_jx_i\) where the scalars \(\theta_{ij}\) are nonzero and depend only on \(i\) and \(j\) (and of course on \(A\)). The matrix \(M^A =(\theta_{ij})\) of order \(r\) is the matrix of commutation relations for the given regular decomposition of \(A\). Regular decompositions were introduced by \textit{A. Regev} and \textit{T. Seeman} [J. Algebra 291, No. 1, 274-296 (2005; Zbl 1083.16017)]. In that paper these decompositions were used in order to study graded tensor products of PI algebras and the corresponding graded identities. In the present paper the authors continue the study and applications of the regular decompositions. They introduce the so-called minimal decomposition of \(A\). They prove (see Proposition 2.1 in the paper) that if \(A\) and \(B\) admit regular decompositions then their tensor product \(A\otimes B\) also does, and if the decompositions of \(A\) and \(B\) are minimal then so is the induced decomposition on \(A\otimes B\). As a corollary the authors obtain that the algebra \(M_n(G)\) is regular (here \(G\) is the infinite dimensional Grassmann algebra). The authors give the interesting Conjecture 2.5: Is it true that a regular decomposition is minimal if and only if the determinant of the corresponding matrix is nonzero? Also, is the number of summands in a minimal decomposition an invariant of the given algebra? Further on the authors study twisted tensor products of graded algebras where the twist is given by a bicharacter on the grading group. They generalize one of the key results of the above cited paper, as well as of the paper by \textit{J. A. Freitas} and the reviewer, [J. Algebra 321, No. 2, 667-681 (2009; Zbl 1172.16013)]. Namely they prove that if \(A\) and \(B\) are \(H\)-comodule algebras where \(H\) is a finite-dimensional bialgebra, and if \(A\) and \(B\) are PI then their twisted tensor product is also PI.