Can I add elements of a superalgebra?

Question

Let's consider a superalgebra $A$ over the real numbers $\mathbb R$. Wikipedia defines this superalgebra via the direct sum: $$ A = A_0 \oplus A_1, \tag{1} $$ and a bilinear multiplication (indices mod 2): $$ A_i A_j \subseteq A_{i+j}. $$ An element $x\in A_i$ has parity $|x|=i$. If we let $x,y\in A_0$ and $\theta,\eta\in A_1$, then: $$ xy=yx, \quad \theta\eta =-\eta\theta, \quad x\theta = \theta x. \tag{3} $$ My question is: can we take the sum of two elements belonging to different $A_i$? In other words, does "$x + \theta$" make sense?

Answer 1

Yes, it does. If $x\in A_0$ and $\theta\in A_1$, then $x+\theta\in A_0\oplus A_1$. If $x+\theta$ didn't exist, $A$ wouldn't even be a vector space, and much less an algebra.