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remark.  [ca-001I]
✍️source

    The definition above is equivalent to the following definition in literature (e.g. [jadczyk2019notes, 1.7]):

    Let \(M\) be a module over \(R\). An algebra \(T\) is called a tensor algebra over \(M\) (or "of \(M\)") if it satisfies the following universal properties:

    1. \(T\) is an algebra containing \(M\) as a submodule, and it is generated by \(M\),
    2. Every linear mapping \(\lambda \) of \(M\) into an algebra \(A\) over \(R\), can be extended to
      3. a homomorphism \(\theta \) of \(T\) into \(A\).