Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$?

Question

Is $\Lambda(T^{*}E)=\bigoplus_{k=0}^n\Lambda^k(T^{*}E)$ a complex line bundle over $T^{*}E$? I know that $\Lambda(T^{*}E)=\bigoplus\Lambda^k(T^{*}E)=\Lambda^0(T^{*}E)\oplus\cdots\oplus\Lambda^n(T^{*}E)$. Since $\Lambda^n(T^{*}E)$ is a line bundle over $T^{*}E$, I guess my question can be stated in a more general way: if $A_1\oplus\cdots\oplus A_n$ is a direct sum of algebras, such that $A_n$ is a line bundle over $X$, then is $A_1\oplus\cdots\oplus A_n$ a line bundle over $X$?