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.
Algebras have two operations by definition and the one thing they have in common is that the multiplication distributes over addition.
Yes, there is no notion of inverses without an identity, the definition of an inverse is in terms of an identity.
Stop posting.
Do you think a group isn’t an algebra? What, by your definitions make an “Algebra” different from a “Ring”?
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.