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For an A-module MA, we prove that M is a finitely generated projective -module if and only if M is a -projective finite-dimensional module, and either M is a reduced module or A is a simple Artinian ring.
      
It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).
      
The problem of whether a given ring R has a left unit was reduced earlier by the author to the semiprime case, namely, R has a left unit if and only if r ∈ Rr for any element r of the prime radical P(R) and the ring RP(R) has a left unit.
      
It is proved that a polynomial in several Mahler measures with positive rational coefficients is equal to an integer if and only if all these Mahler measures are integers.
      
It is proved that the semigroup of all triangular n × n matrices over a finite field K is inherently nonfinitely based if and only if n >amp;gt; 3 and |K|>amp;gt; 2.
      
It is proved that M admits a Hausdorff module topology preceding the box topology in the lattice of all module topologies if and only if the dimension of the vector space M over R is a measurable cardinal.
      
It is proved that a generic-type 6-dimensional almost Hermitian submanifold of the algebra of octaves is minimal if and only if it belongs to the Gray--Hervella class G2.
      
A right distributive ring A is a right order in a right uniserial ring if and only if the set N(A) is a left ideal of A.
      
In particular, it is proved that the Olevskii system forms an RUC basis in Lp[0,1] if and only if 2 ≤ p >amp;lt; ∞.
      
With the help of this result, it is proved that a free product with amalgamated subgroups of two finitely generated Abelian groups is a residually finite p-group if and only if it is conjugacy p-separable.
      
An Abelian group A is Lw1, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.
      
An Abelian group A is Lw1, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.
      
The Yang-Mills equation with coefficients in a Lie algebra L has nontrivial solutions with constant coefficients if and only if the Lie algebra L is not antinilpotent.
      
A Compact Space Is Homotopy Equivalent to a CW-Complex If and Only If It Is an Absolute Neighborhood h-Retract
      
It is proved that a compact space is homotopy equivalent to a CW-complex if and only if it is an absolute neighborhood h-retract.
      
We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero.
      
We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero.
      
It is proved that a regular locally conformally quasi-Sasakian structure is normal if and only if it is locally conformally cosymplectic and has closed contact form.
      
It is shown that the Kenmotsu structures have these properties and that a structure with the above properties is a Kenmotsu structure if and only if its contact Lee form coincides with the contact form.
      
A lattice of right divisors of a linear ordinary differential operator P is distributive if and only if all factors of the operator P are not parameterized.
      
 

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