RSA laboratories have established the leading standards for RSA encryption in the PKCS#1 standard. I would not say that the algorithm is completely separate from the company. That said, the schemes within PKCS#1 have been presented in scientific papers extending all the way back to the original proposal by Rivest, Shamir and Adleman (that make up RSA the acronym).
The values of the keys however are independent of RSA Security / Laboratories / EMC / Dell (there were a few takeovers). The values $p$, $q$ and $n$ are different for each RSA key pair (if the RSA KeyGen algorithm is followed correctly and a secure random number generator is used). Generation of these values should take place in a location under control of the entity requiring the key pair. Only the public key - consisting of the pair $(n, e)$ - is then distributed.
Related is the Kerckhoff's principle. The algorithm should be secure as long as the keys are secure. The RSA algorithm including the PKCS#1 padding schemes has had extensive analysis from cryptographers.
So no, you do not have to trust RSA - the company - with your encryption. You can of course if you want to; trusting a specialized security company is often more secure than trying to implement it yourself. However if you still want to use RSA - the company - for that is up to debate to say the least.