A group is not an algebra. A group consists of a single associative binary operation with an identity element and inverses for each element.
A ring is an abelian (commutative) group under addition, along with an additional associative binary operation (multiplication) that distributes over addition. The additive identity is called zero.
A field is a ring in which every nonzero element has a multiplicative inverse.
A vector space over a field consists of an abelian group (the vectors) together with scalar multiplication by elements of the field, satisfying distributivity and compatibility conditions.
A non-associative algebra is a vector space equipped with a bilinear multiplication operation that distributes over vector addition and is compatible with scalar multiplication.
An (associative) algebra is a non-associative algebra whose multiplication operation is associative.
You can read more about these definitions online and in textbooks - these are standard definitions. If you are using different definitions, then it would help your case to provide them so we can better understand your claims.
A group is not an algebra. A group consists of a single associative binary operation with an identity element and inverses for each element.
A ring is an abelian (commutative) group under addition, along with an additional associative binary operation (multiplication) that distributes over addition. The additive identity is called zero.
A field is a ring in which every nonzero element has a multiplicative inverse.
A vector space over a field consists of an abelian group (the vectors) together with scalar multiplication by elements of the field, satisfying distributivity and compatibility conditions.
A non-associative algebra is a vector space equipped with a bilinear multiplication operation that distributes over vector addition and is compatible with scalar multiplication.
An (associative) algebra is a non-associative algebra whose multiplication operation is associative.
You can read more about these definitions online and in textbooks - these are standard definitions. If you are using different definitions, then it would help your case to provide them so we can better understand your claims.