# Adobe Photoshop 2021 (Version 22.3) KeyGenerator [Win/Mac]

## Adobe Photoshop 2021 (Version 22.3) Keygen

Q: Associative products when dealing with modules/rings/algebras Consider the following definition of a module $M$ over a ring $R$: A module $M$ over a ring $R$ is an abelian group $M$ with a ring homomorphism $\phi : R \to \text{End}(M)$ into the endomorphisms of $M$. We can give $M$ a multiplication by associating to any $a,b \in R$ a map $$M \times M \to M$$ given by $$(x,y) \to x \cdot y := \phi(a) \circ \phi(b) \circ x \circ y$$ Is this associativity well defined? I guess so since any other way would have to preserve the identity element $e_M$. This is a strange to me definition since this does not make sure that products are preserved by some other “magic” identity, e.g. using the inverse in the ring $R$. I guess this is the reason why they introduced this definition by associating a ring homomorphism. So my question is: Is this definition of modules associative? If yes, why? If not, what is a better definition to make sure products are well defined? A: Let $R$ be a unital ring and $M$ be a unital $R$-module. Let $\phi:R\to\text{End}(M)$ be the map associating to $r\in R$ the left multiplication by $r$. It is a ring homomorphism because $(1r)(sm) = (rs)m = rs(m) = (rs)m$ for all $r,s\in R$ and $m\in M$. Now we can construct a left multiplication $R\times R\to R$ as follows: $$(r_1,r_2)\mapsto r_1\phi(r_2)$$ This map is well-defined since: (rs)\phi(t) = (rs)e_M = e_M(r’s) = e_M\phi(r’s) = (r’s)\phi(t) = r_1\phi(r_2)\phi(