After the idea arrives to the senate, who is it assigned to

If you are talking about a bill, it is debated with the House of Representatives, and then sent to the President for signature.

If you mean bills, Generally, bills are legislative proposals that, if enacted, carry the force of law, whereas resolutions do not. Though, this is not always true ... A Bill starts either in the House of Representatives or in the Senate. Bills that start in the House are designated "H. R."