My ex doesn't work at all and has a new husband that doesn't either and I pay child support year round, can I claim my children because I'm the main provider?

You can only claim them if your ex releases the exemption to you on form 8332 (assuming they live with your ex more than half the year). If she does, you can only claim the dependency exemption and child tax credit, not the dependent care credit or EIC.

Is this issue addressed in the divorce decree?

In general the answer is no - only the custodial parent can claim the child as a dependent regardless of who pays the bills. The only exception is if the custodial parent is willing to relinquish that right by giving you a signed form 8332 to include with your tax filing, as noted earlier by TfA. She must provide that form to you for each year that you claim the child as your dependent.

All that ebaines and taxforaliens says is true, but savvy divorce attorneys usually include a provision that calls for the parents to split claiming the exemptions on alternating years.

Suggest you check your divorce decree, which the IRS WILL abide by under the law.