Let \({\mathcal {A}}\) and \({\mathcal {U}}\) be Banach algebras such that \({\mathcal {U}}\) is a Banach \({\mathcal {A}}\)-bimodule with compatible algebra multiplication, module actions and norm. By defining an appropriate action, we turn \(\ell ^1\)-direct product \({\mathcal {A}}\times {\mathcal {U}}\) into a Banach algebra such that \({\mathcal {A}}\) is a closed subalgebra and \({\mathcal {U}}\) is a closed ideal of \({\mathcal {A}}\times {\mathcal {U}}\). This algebra is, in fact, the semidirect product of \({\mathcal {A}}\) and \({\mathcal {U}}\), \({\mathcal {A}}\ltimes {\mathcal {U}}\), and as well as, every semidirect product of Banach algebras can be represented as this form. Our aim in this paper is to study the first cohomology group of \({\mathcal {A}}\ltimes {\mathcal {U}}\) and investigate the relation between the first cohomology group of \({\mathcal {A}}\ltimes {\mathcal {U}}\) and those of \({\mathcal {A}}\) and \({\mathcal {U}}\). As an application of our results, we show that the earlier results obtained for the first cohomology group of various classes of Banach algebras such as direct products of Banach algebras, module extension Banach algebras and \(\theta\)-Lau products of Banach algebras can be obtained directly by applying our main results and techniques of proofs. Some examples are also given.

令\（{\ mathcal {A}} \）和\（{\ mathcal {U}} \）是Banach代数，使得\（{\ mathcal {U}} \}是Banach \（{\ mathcal {A }} \）- bimodule具有兼容的代数乘法，模块动作和范数。通过定义适当的动作，我们将\（\ ell ^ 1 \）正乘积\（{\ mathcal {A}} \ times {\ mathcal {U}} \）变成Banach代数，使得\（{\ mathcal {A}} \）是一个封闭的子代数，\（{\ mathcal {U}} \}是\（{\ mathcal {A}} \ times {\ mathcal {U}} \}的封闭理想。实际上，该代数是\（{\ mathcal {A}} \）和\（{\ mathcal {U}} \\}的半直接乘积，\（{\ mathcal {A}} \ ltimes {\ mathcal {U}} \）以及Banach代数的每个半直接乘积都可以表示为这种形式。本文的目的是研究\（{\ mathcal {A}} \ ltimes {\ mathcal {U}} \）的第一个同调群，并研究\（{\ mathcal {A }} \ l次{\ mathcal {U}} \）以及\（{\ mathcal {A}} \）和\（{\ mathcal {U}} \}的时间。作为我们结果的应用，我们显示了从第一类同种Banach代数的同调子组获得的较早结果，例如Banach代数的直接乘积，模块扩展Banach代数和\（\ theta \）-通过应用我们的主要结果和证明技术，可以直接获得Banach代数的Lau产品。还给出了一些例子。