No, that is wrong. The mRNA can't pass through the cell's nuclear membrane. It can't interact with the cell's DNA in any way. It is our own immune cells which use the mRNA to develop a sensitivity to the spike protein which would be present on the surface of a live virus. Once the immune cell has created that protein the mRNA degrades into nucleotides that are absorbed by the immune cell.
The inert string of mRNA can't duplicate itself. Unlike the mRNA released by a live virus which is capable of duplicating itself by creating exponentially increasing copies of the virus itself.
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