What is the meaning of 'homogeneous' here? And what does it mean by 'degree'? This is a part extracted from a textbook that has many definitions that I was confused and failed to find.

Let $\displaystyle A=\oplus_{n=0}^\infty A_n$ be a Noetherian graded ring. Then $A_0$ is a Noetherian ring and $A$ is generated (as an $A_0$- algebra) by say $x_1,...,x_s$ which we may take homogeneous, of degrees $k_1,...,k_s$(all $>0$).

My question is what is the meaning of 'homogeneous' here? And what does it mean by 'degree'? The text book does not elaborate it anywhere and I could not find a clear definition from any source. Even Wikipedia gives a very confusing definition. Could somebody please help me to understand this?
 A: So when you have a graded ring $A = \oplus_{n \geq 0} A_n$, an element of $A_n$ is called a homogeneous element. If the element is non-zero it has degree $n$. 
People have different conventions about what the degree of $0$ is (it lives in all the $A_n$'s). 
Note that when you write $A$ as a direct sum of abelian groups, you are implicitly assuming that you cannot have a nonzero element which is simultaneously living in $A_n$ and $A_m$ for $n \neq m$. So degree of a non-zero element is well defined.
If you are confused, just think about the example of a polynomial ring say of one variable. If you are familiar with homogenous polynomials, and what the degree of a non-zero homogeneous polynomial is, then the leap to an arbitrary graded ring $A = \oplus_{n \geq 0} A_n$ is really no different.
However, while in the case of a polynomial ring we often talk about degree of a polynomial which may be a sum of monomials of different degrees, if you come across the phrase ''$x \in A = \oplus_{n \geq 0} A_n$ has degree $m$'', it usually means $x$ is homogeneous to begin with even if it is not mentioned explicitly that $x$ is a homogeneous element. Hope this helps.
