ABSTRACT
Let X be a countable set and let F(X) be the free associative algebra generated by X over an infinite field F. The T-ideal https://www.w3.org/1998/Math/MathML"> Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math587.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the algebra https://www.w3.org/1998/Math/MathML"> F 〈 X 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math588.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called verbally prime if for every T-ideals https://www.w3.org/1998/Math/MathML"> Γ 1 , Γ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math589.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the inclusion https://www.w3.org/1998/Math/MathML"> Γ 1 Γ 2 ⊆ Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math590.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> implies one of the inclusions https://www.w3.org/1998/Math/MathML"> Γ 1 ⊆ Γ or Γ 2 ⊆ Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math591.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The T-ideal https://www.w3.org/1998/Math/MathML"> Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math592.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called verbally semiprime if there are no non-trivial T-ideals nilpotent modulo https://www.w3.org/1998/Math/MathML"> Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline-math593.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . A variety of the algebras is called prime (semiprime) if the ideal of its polynomial identities is verbally prime (semiprime).
