user5523: I've understand that the on-chain code is stored in the blockchain through Playground module which is in sync against a local node. But how to "publish" this code to run anywhere? It's an accessible process, It must have some "certification"?

user4596: First of all on-chain code isn't stored in the blockchain, the hash of on-chain code is.
To use on-chain validator on testnet or mainnet steps are:

Compile on-chain code into plutus script
Generate address from plutus script
Lock funds into plutus script address (Don't forget Datum!!!)
Pass plutus script for validation to unlock funds from plutus script address

You can try going trough Alonzo exercises for an example how to do all of these steps.
And as for certification I don't think there is an official available certification yet

5523: <p>I've understand that the on-chain code is stored in the blockchain through Playground module which is in sync against a local node. But how to &quot;publish&quot; this code to run anywhere? It's an accessible process, It must have some &quot;certification&quot;?</p>
 4596: <p>First of all on-chain code isn't stored in the blockchain, the hash of on-chain code is.</p>
<p>To use on-chain validator on testnet or mainnet steps are:</p>
<ol>
<li>Compile on-chain code into plutus script</li>
<li>Generate address from plutus script</li>
<li>Lock funds into plutus script address (<a href="https://github.com/input-output-hk/cardano-node/issues/3069" rel="nofollow noreferrer">Don't forget Datum!!!</a>)</li>
<li>Pass plutus script for validation to unlock funds from plutus script address</li>
</ol>
<p>You can try going trough <a href="https://github.com/input-output-hk/Alonzo-testnet/blob/main/Alonzo-solutions/exercise3/e3SampleSolution.md" rel="nofollow noreferrer">Alonzo exercises</a> for an example how to do all of these steps.</p>
<p>And as for certification I don't think there is an official available certification <a href="https://iohk.io/en/blog/posts/2021/10/25/new-certification-levels-for-smart-contracts-on-cardano/" rel="nofollow noreferrer">yet</a></p>
